Download Review of Functions and Transformations

Math 1 EOCT Review
Name
Review of Functions and Transformations
Terms/Concepts You Need to Know - coordinate plane, the quadrants, the x-axis, the y-axis, the origin, xcoordinate, y-coordinate, relations, functions, vertical line test, domain, range, mapping, graph, order of
operations, how to evaluate a function, linear functions, coefficients, the parent functions, transformations of the
parent functions
1.
2.
Label the quadrants, axes, and origin on the
coordinate plane to the right.
4
2
Plot the points on the coordinate plane.
A (0, 3) B (-2, 4) C (1, -5) D (-3, -1)
-5
5
-2
-4
-6
--------------------------------------------------------------------------------------------------------------------2
For Questions 3-7, evaluate if f ( x)  3x  1 , g ( x)  x 2  8 and h( x) 
.
x3
3. f (2)
4. g (1)
5. h(3)
6. g ( 3)
7. h(3)
--------------------------------------------------------------------------------------------------------------------For Questions 8-14, is the relation a function?
8.
9.
10.
11.
4
4
4
4
2
2
2
2
-5
5
-5
5
-5
5
-5
5
-2
-2
-2
-2
-4
-4
-4
-4
-6
-6
-6
-6
12.
{(2,3), (2, 4), (7,10)}
13.
{(3, 0), (4, 0), (5, 0), (6,1)}
x
1
2
3
4
14.
y
1
2
3
4
--------------------------------------------------------------------------------------------------------------------For Questions 15-21, name the parent function of the function whose equation is given.
15. j ( x)  x  1
1
19. n( x)   x3  11
4
16. k ( x) 
2
x 1
5
20. p ( x)  400 x
17. l ( x) 
7 2
x
3
21. q ( x)  
18. m( x)  
2
x7
13
9
8
2x
22.
True or False: All functions are relations.
23.
In what quadrant is the point whose coordinates are (-3, 9)?
24.
On which axis is the point whose coordinates are (0, 1)?
For Questions 25-32, describe the transformation in relation to the parent function.
25. r ( x) 
2
x
5
29. v( x)  9 x
28. u ( x ) 
26. s ( x)   x  1
27. t ( x)  10 x 2  9
30. w( x)  x  12
1
31. b( x)   x  1.3
5
9 3
x
2
4
32. c( x)   x
3
--------------------------------------------------------------------------------------------------------------------For Questions 33-38, match the equation for the function on the left with its graph on the right.
d ( x)   x 2  3
33.
A)
f ( x)  4 x 2  3
34.
B)
4
4
4
2
2
2
-5
g ( x)   x 2  3
35.
1
h( x )   x 2  3
4
36.
k ( x)  4 x 2  3
5
-5
5
-2
-2
-4
-4
-4
-6
-6
-6
E)
F)
4
4
4
2
2
2
-5
38.
-5
5
-2
D)
j ( x)  x 2  3
37.
C)
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
--------------------------------------------------------------------------------------------------------------------For Questions 39-41, name the parent function, and describe the transformation in relation to the parent
function.
39.
40.
41.
4
4
4
2
2
2
-5
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
Review on Function Characteristics
Name_________________________
For the graph of the function, find the following: domain, range, intervals of increase and decrease, maximum,
minimum, x and y-intercepts, and solutions (or zeros). In addition, state whether the function is “even”, “odd”,
or “neither”. Do not find the intervals of increase and decrease for #5.
1.
2.
3.
6
6
6
4
4
4
2
2
2
-5
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
4.
5.
6
6
4
4
2
2
-5
5
6.
-5
5
-2
-2
-4
-4
-6
-6
7.
8.
6
6
6
4
4
4
2
2
2
-5
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
9.
On the front side, which of the functions graphed are discrete? Which are continuous?
Question 6 is neither discrete nor continuous.
-------------------------------------------------------------------------------------------------------------------For Questions 10-15, state if the function whose equation is given is “even”, “odd”, or “neither”.
10. f ( x)  8 x
11. g ( x)  x 2  9
12. h( x)  x 4
13. j ( x)   x  3
14. k ( x)  x  1
15. l ( x)  x
--------------------------------------------------------------------------------------------------------------------For Questions 16-18, state the x and y intercepts for the function whose equation is given.
16. y  5 x
17. y  2 x  1
18. y  4 x  20
19.
How does one find the solutions or zeros of a function using the equation like in
Questions 16-18?
--------------------------------------------------------------------------------------------------------------------For Questions 20-23, sketch the graphs of the functions described below. Describe the domain and range for
Questions 20, 21, and 23.
20. The output is the number of letters
in the name of the number inputted.
The domain is the whole numbers from
zero to ten.
21. The height above the ground of a water
balloon thrown high into the air is a
function of the time since it was thrown.
22. The profit for a coffee shop in Idaho
is a function of the amount charged for
a regular cup of coffee.
23. The length of a pencil is a function of
the time since it was removed from its
box to be used.
Sequence and Rate of Change Review
Name
4
For questions 1- 2, use the graph at the right.
1. What is the rate of change between x = -6 and x = -3?
2
2. What type of rate of change is shown in this in this graph?
-5
-2
-4
--------------------------------------------------------------------------------------------------------------------For questions 3- 6, use the definition
.
3. What type of definition is this?
4. What is
?
5. What is
?
5. What would the first term be for this sequence?
6. What is the rate of change for the sequence this equation produces?
-------------------------------------------------------------------------------------------------------------------Use the table to answer questions 7-8.
x
-5 -2 1 5
f(x) 7
4
1 -5
7. What is the rate of change from x = -2 and x = 5?
8. Does this table as a whole have a constant or variable rate of change?
--------------------------------------------------------------------------------------------------------------------Use the equation
for questions 9-10.
9. What are the third, forth, and fifth terms?
10. What is the domain and range for the first 5 terms?
--------------------------------------------------------------------------------------------------------------------In the movie Edward Scissorshands, Edward started using his scissor hands to hair but the first 4 days of his
‘new occupation’ he only cut 8 peoples hair. However, as his popularity grew, more people went to him for a
haircut as seen through the fact that the rest of that week, he was booked, cutting 17 peoples hair. (Assume he
worked every day that week and remember how many days are in a week)
11. What is the rate of change for the first 4 days?
12. What is the rate of change for the rest of the week?
13. What is the rate of change for the whole week?
--------------------------------------------------------------------------------------------------------------------Use the pattern to answer questions 14-18. The pattern represents a sequence of numbers.
14.
Draw the next two figures.
15.
Write the domain of the shown sequence.
16.
Write a function to calculate the nth term of the sequence. (Explict/Closed form)
17.
Write a recursive formula for the sequence.
18.
Use your function to determine the number of dots in the 12th figure.
--------------------------------------------------------------------------------------------------------------------Write the closed form definition for each sequence.
19. 3, 1, -1, -3,…
20. 4, 11, 18, 25,…
--------------------------------------------------------------------------------------------------------------------Write a recursive formula for each sequence.
21. 2, -6, 18, …
22. -3, -11, -43, -173…
--------------------------------------------------------------------------------------------------------------------23.
c. This chart shows what kind of rate of change?
--------------------------------------------------------------------------------------------------------------------24. What does mean?
25. What does
represent?
26. What is the formula you must remember in order to write the closed form definition?
 This definition is also known as the explicit definition
Logic Test Review

Name
Tell if the statements use deductive or inductive reasoning:
◊
Gravity makes things fall. Therefore, the apple that hit Johnny Appleseed’s head must have been due
to gravity.
◊
As I was leaving Regal Cinemas, I noticed that everyone walking in had umbrellas. I turned to my
friend and said, “Great! It must have started raining while we were in the movie!”
◊
All dogs bark. The animal in my backyard must be a dog because it barks.
◊
My math teacher told me that a2 + b2 = c2. So if I am given a= 3 and b= 4, then c= 5.
◊
According to the advertisement, Chevrolet makes reliable cars. Therefore, because I drive a Chevrolet
Tahoe, I assume that I am not going to need roadside assistance on my insurance because I’m not
going to break down.
◊
My Tahoe is black. The trail blazer parked beside me is black. All SUV’s are black.
◊
If my dad is behind a bad driver, he always makes a comment that it must be a woman behind the
wheel. Having grown up hearing those statements, my brother sees a bad driver and automatically
assumes that a woman is driving the car.
◊
After getting clawed and bitten by cats multiple times when I was little, I have developed a fear of cats
thinking that all cats are mean and vicious.
◊
Someone once told me, “This picture was taken at the ONLY place worth going: Hilton Head.” So
when I heard my friend talking about a great trip she had taken, I said, “Oh, so you went to Hilton
Head?”
Put the statement in if- then form; tell if it is true or false ( if they are false, give a counterexample); then state
the inverse.
Sundays are always rainy.
a. If- then: ___________________________________________
b. T / F counterexample ________________________________
c. Inverse: ____________________________________________
One would have to go to Evans Cinemas if they want to see a movie.
d. If- then: ___________________________________________
e. T / F counterexample ________________________________
f.
Inverse: ____________________________________________
State the inverse, converse and contrapositive of the statements
If the animal is a wolf, then it has sharp teeth.
g. Inverse: _____________________________________
h. Converse: _________________________________________
i.
Contrapositive: ______________________________________
If the animal is the mascot for LHS, then it is a panther.
j. Inverse: _____________________________________
k. Converse: _________________________________________
l.
Contrapositive: ______________________________________
What is the converse of the statement?
a)
b)
c)
d)
If it tastes good, then it is pizza.
If it is not pizza, then it does not taste good.
If it is pizza, then it takes good.
If it does not taste good, then it is not pizza.
If it is pizza, then it does not taste good.
What is the conclusion in the statement?
a)
b)
c)
d)
You need an umbrella if it is raining.
It rains
You need an umbrella
If you need an umbrella, then it rains.
If it does not rain, then you do not need an umbrella.
What is the inverse of the statement?
a)
b)
c)
d)
If you like the color green, then you like apples.
If you do not like the color green, then you do not like apples.
If you like apples, then you do not like the color green.
If you do not like apples, then you do not like the color green.
You like grapes.
What is the contrapositive of the statement?
a)
b)
c)
d)
If it is dark outside, then it is night time.
If it is night time, then it is dark outside.
If it is not dark outside, then it is not night time.
If it is not night time, then it is not dark outside.
If it is not night time, then it is dark outside.
What is the hypothesis in the statement?
a)
b)
c)
d)
If he is a boy, then his name is George.
He is a boy
His name is George
If his name is George, then he is a boy.
If he is not a boy, then his name is not George.
What is the inverse of the statement?
a)
b)
c)
d)
If it is Wednesday, then it is not the weekend.
If it is not Wednesday, then it is not the weekend.
If it is not the weekend, then it is Wednesday.
If it is the weekend, then it is not Wednesday.
If it is not Wednesday, then it is the weekend.
Name
Michael went to the barber shop to get his haircut. As he walked in he
saw someone with blue hair walking out. When he walked in the shop
the cashier had pink hair. He decided to leave and go to a different
barber shop because he thought that everyone who got their haircut at
that shop came out with unusual hair colors. He applied which of the following?
a)
b)
c)
d)
e)
Logical reasoning
Inductive reasoning
Deductive reasoning
Superficial reasoning
None of the above
What would be the if-then form of the statement?
_________
Trix are for kids.
a) If you are a silly rabbit, then Trix are for kids.
b) If the cereal is Trix, then it is for kids.
c) If it is for kids, then the cereal is Trix.
d) If you are a kid, then Trix are for you!
e) None of the above
What type of statement has the same truth value as the contrapositive?
a. Inverse
b. Converse
c. Conditional
d. Conclusion
Provide a counterexample for the false statement:
If a number ends in 0, then it is divisible by 20.
______________________________________________________________________
If it is the name of a state in the United States, then it does not start with the letter I.
If it is an odd number, then it is prime.
________________________________________________________________________
Review for Operations with Polynomials
Simplify completely. Your answer cannot have negative exponents. Assume all variables do not equal zero. If
an expression cannot be simplified, write “cannot be simplified”.
1.
17.
2.
18.
3.
19.
4.
20.
5.
21.
6.
22. (
7.
23.
24.
8.
9.
10.
11.
12.
•
13.
•
14.
•
15.
16.
•
+ mn •
•
n•
Name ________________________
25.
28.
31.
26.
29.
32.
27.
30.
33.
34. A square has an unknown side length. The length of the
square is extended by 5 inches, and the width is extended
by 11 inches to form a new rectangle. In the space to the
right, draw a figure which represents this situation.
35. Write a simplified algebraic expression for the area of the new
rectangle formed in question 34.
36. Create a geometric representation with the following information:
A rectangle has an unknown width.
Its length is 4 times is width.
Both the length and width are extended by 3 units.
Write a simplified expression for its perimeter.
37. Find the area of a rectangle with the sides that are 3x-1 and 4x+2 in length?
38. Suppose a fence has a length of 100 yards and it is going to be extended by an unknown number
yards. Write an algebraic expression that represents the length of the new fence.
Radical Expressions Review
Simplify completely:


1.
63x 5
2. 2  7 2  7
5.
9
8
6. 4 63  7
9. 16 x 3  4 x 2 y 3
10.
4 5
x
13. 5 3  7 3
14.
17. 9 11  3 6  4 54  12 44
20.
24.
6 x 4 25 x 3
2 x 16 y 3
32
49
21.

3
31
36
18.
3.
24
3 16x
4. 11  5 32
7.
9x
3x
8. 12  27
11. 2 98
12.
15. 3 6  2 54
16.
12
4
9 x 10 y 8 z 13
25. 3 2  7 24  4 2  5 24
8 8x
5 12 x 7
7
5 18
19. 5 2  3 5
22.
5 x 2a 3
6 x 2a 5
26. a3c a 4b19c3
23.
45
27.
x4
Review: Factoring
Name
Factor the greatest common factor out of the polynomial. If the GCF is 1,
write PRIME.
1. x3 + x2 + x
2. 15a + 12b + 6c
3. 8x2 + -18y2
4. x2y + 2y
5. 64c3 + -56c2 + 88c
6. 18k + 36k2 + 9k3
7. a3b2 + a3b4 + ab4
8. 6q + 10q2 + 8q3
9. 14dj + 14ej
10. m3n2 + -1m2n3
11. -44c2 - 48c + 80
12. -4x2 + 4x + 63
Factor the expression. If nothing can be factored, write PRIME.
13. 12 p3  21 p 2  28 p  49
14. 25v3  5v 2  30v  6
15. b 2  8b  7
16. b 2  6b  8
17. 5n 2  10n  20
18. 16n 2  9
19. 21k 3  84k 2  15k  60
20. 9 x 2  1
21. mz  5mh 2  5nz  25nh 2
22. x 6  5 x 4  7
23. x 4  2 x 2  15
24. 225  4x 2
25. x 2  3 x  10
26. x 2  9 x
27. 2 x 2  8
28. x 2 y  5xy  24 y
29. x 2  9 x  16
30. x 4  10 x 2  25
31. x 2  15 x  16
32. 4 x 2  64
33. 2k 2  22k  60
34. 24r 3  64r 2  21r  56
35. x 6  2 x 4  16 x 2  32
36. 12 xy  28 x  15 y  35
37. 49  c 2
38. x 2  y 2
39. k 4  36
40. x 2  20 x  100
Review for Triangles, Quads, & Polygons (1)
Name_________________________
For Questions 1-6, can a triangle be formed with the given side lengths? Suppose that all of the
given measurements are in centimeters.
1. 13, 15, 28
2. 10, 10, 18
3. 7, 7, 7
4. 1.3, 2.5, 4.8
5. 1, 99, 101
6. 22, 32, 32
--------------------------------------------------------------------------------------------------------------------7.
Triangle ABC has interior angles such that mA  8x 1 , mB  7x 10 , and
mC  6x  2 . What is the value of x?
--------------------------------------------------------------------------------------------------------------------8.
Fill-in-the-blanks: The diagonals of a kite are _____________, while the diagonals of
an isosceles trapezoid are _________________.
J
9.
M
24
18
In the figure to the left, JMLK is a kite such that JK  JM . If
JN  24 , and KN  18 , what is JK ?
N
K
L
--------------------------------------------------------------------------------------------------------------------10.
Suppose that DEF has the following side lengths: DF  29 inches, EF  24.5 inches,
and DE  27.63 inches. List the interior angles in order from the angle with the smallest
measure to the angle with the largest measure.
--------------------------------------------------------------------------------------------------------------------For Questions 11-15, write the correct word in the blank that describes all parallelograms.
11.
The _______________ bisect each other.
12.
Opposite angles are _________________.
13.
_______________ sides are congruent.
14.
Consecutive angles are _________________.
15.
Opposite ______________ are parallel.
----------------------You must memorize these statements about parallelograms.-----------------16.
Suppose that a trapezoid has interior angles which measure 113 , 101 , and 65 . What
is the measure of the fourth interior angle?
---------------------------------------------------------------------------------------------------------------------
17.
STUV (pictured to the right) is a rhombus. If mV  121 ,
and mU  4 y  3 , what is the value of y?
S
V
T
121
4y + 3
U
--------------------------------------------------------------------------------------------------------------------18.
This is a question from last year's end-ofcourse test. Using the diagram to the left,
what is the mQPR ? Make sure you
answer what the question is asking!
--------------------------------------------------------------------------------------------------------------------19.
Fill-in-the-blanks: The measure of an exterior angle of a triangle is equal to the sum of
the measures of the two _____________ _____________ angles.
20.
What is the sum of the measures of the exterior angles, one at each vertex, of a
convex 14-gon?
--------------------------------------------------------------------------------------------------------------------21.
112
94
129
The measures of the angles given in the polygon to
the left are the measures of the interior angles of the
polygon. What is the measure of the interior angle
that is not provided in the diagram?
107
--------------------------------------------------------------------------------------------------------------------22.
Chelsea is cutting pieces of string into different lengths that are a whole number of
inches. If two pieces measure nine inches and fourteen inches in length, what are
the shortest and longest lengths possible for the third string if she wishes to form a
triangle with the three pieces?
23.
Consider triangle KLM. There is an exterior angle located at vertex M that
measures 146 . If K and L have the same measure, then what is mL ?
--------------------------------------------------------------------------------------------------------------------24.
What is the measure of one interior angle for a regular 11-gon? Round your answer
to the nearest tenth (one decimal place).
Name ________________________
--------------------------------------------------------------------------------------------------------------------For Questions 25-28, find the requested value for the problem involving a quadrilateral.
25.
If an interior angle of a rectangle
measures 10 z  6 , what is the
value of z?
26.
In rhombus ABCD below, EC  48 ,
and DC  50 . What is ED?
B
E
A
48
C
50
D
27.
An isosceles trapezoid has one diagonal
with a length of 77 centimeters, and the
other diagonal has a length of 4a  9
centimeters. What is the value of a?
28.
Parallelogram HOSA has interior
angles such that mH  4b , and
mS  6b  34 . What is the value
of b?
--------------------------------------------------------------------------------------------------------------------29.
Each interior angle inside of a regular polygon measures 165 . How many sides does
the polygon have?
--------------------------------------------------------------------------------------------------------------------X
W
30. The figure to the left is a parallelogram. If the
length of ZX is 104 meters, and ZU  11c  3 meters,
what is the value of c?
U
Z
Y
--------------------------------------------------------------------------------------------------------------------31.
What is the measure of each exterior angle of a regular 12-gon?
32.
One side of a rectangle measures four yards. One diagonal of the rectangle measures
five yards. What are the lengths of the other three sides of the rectangle?
--------------------------------------------------------------------------------------------------------------------Consider the figure to the right. Let mQRT  128 ,
and mT  45 .
33.
S
What is mS ?
R
128
34.
What is mTRS ?
45
Q
T
---------------------------------------------------------------------------------------------------------------------
Consider a regular 20-gon.
35.
What is the sum of the measures of its interior angles?
36.
What is the measure of each of its interior angles?
37.
What is the sum of the measures of its exterior angles, one at each vertex?
38.
What is the measure of each exterior angle?
--------------------------------------------------------------------------------------------------------------------For Questions 39-45, is the statement true or false?
39.
In a square, all sides are congruent, and all interior angles are congruent.
40.
When the Pythagorean Theorem is stated as a 2  b 2  c 2 , the quantities labeled as
a and c are the two sides which form the right angle of the right triangle.
41.
All parallelograms are rectangles. (Hint - when you see a statement like this, think
of it as reading, "Parallelograms have all of the characteristics that rectangles have."
42.
All parallelograms are quadrilaterals.
43.
All rectangles are squares.
44.
All squares are rectangles.
45.
The diagonals of a rhombus are congruent.
--------------------------------------------------------------------------------------------------------------------46.
In XYZ , mX  44 , and mY  77 . List the side lengths in order from the side with
the shortest length to the side with longest length.
--------------------------------------------------------------------------------------------------------------------The sum of the measures of the interior angles of a triangle is 180 .
The sum of the measures of the interior angles of a quadrilateral is 360 .
--------------------------------------------------------------------------------------------------------------------In any convex polygon (primarily used in triangles), an exterior angle
and its adjacent interior angle are supplementary.
1 2
m1  m2  180
--------------------------------------------------------------------------------------------------------------------1
2
3
In a triangle, the measure of an exterior angle is equal to the
sum of the measures of the two remote interior angles.
m3  m1  m2
--------------------------------------------------------------------------------------------------------------------The sum of the lengths of the two shortest sides of a triangle are greater than the length of the
longest side. Three segments that do not abide by this rule cannot form a triangle.
--------------------------------------------------------------------------------------------------------------------In a triangle, the angle with the largest measure is opposite the side with the longest length.
Similarly, the angle with the smallest measure is opposite the side with the shortest length.
Name ________________________
The Pythagorean Theorem for Right Triangles: a  b  c . In this formula, the quantities
named a and b are always the lengths of the two sides that make up the right angle. The quantity
named c is the length of the side opposite the right angle.
--------------------------------------------------------------------------------------------------------------------2
2
2
Quadrilaterals
Three types of quadrilaterals are kites, parallelograms,
and trapezoids.
kites
parallelograms
all other
trapezoids quadrilaterals
--------------------------------------------------------------------------------------------------------------------A kite is a quadrilateral that has exactly two distinct pairs of adjacent congruent sides.
The diagonals of a kite are perpendicular - this allows for opportunities to use the Pythagorean
Theorem.
--------------------------------------------------------------------------------------------------------------------The Five Properties of All Parallelograms
1.
The diagonals bisect each other.
2.
Consecutive angles are supplementary.
3.
Opposite angles are congruent.
4.
Opposite sides are parallel.
5.
Opposite sides are congruent.
Parallelograms
rhombuses squares
rectangles
all other
parallelograms
There are three types of parallelograms: rhombuses,
rectangles, and squares. Therefore, these three shapes have
all of the properties of parallelograms!
--------------------------------------------------------------------------------------------------------------------A rhombus is a type of parallelogram in which all four sides are congruent.
The diagonals of a rhombus are perpendicular. Once again, problems involving the Pythagorean
Theorem will involve its diagonals.
Also, the diagonals bisect opposite angles. Essentially, this means that the diagonals cut each
interior angle into two equal pieces.
--------------------------------------------------------------------------------------------------------------------A rectangle is a type of parallelogram in which all of the interior angles are congruent. This
means that each interior angle measures 90 . Again, this opens up opportunities for the
Pythagorean Theorem.
The diagonals of a rectangle are congruent.
--------------------------------------------------------------------------------------------------------------------A square is a type of parallelogram in which all four sides are congruent AND all four interior
angles are congruent. In the figure above, you can see that a square is like a mixture of a
rhombus and a rectangle. This means it has all of the characteristics of both of them.
---------------------------------------------------------------------------------------------------------------------
Finally, completely unrelated to parallelograms, a trapezoid is a quadrilateral in which exactly
one pair of opposite sides are parallel.
A special type of trapezoid is the isosceles trapezoid. An isosceles trapezoid is a trapezoid in
which the pair of non-parallel sides are congruent.
A special property of the isosceles trapezoid (not all trapezoids) - its diagonals are congruent.
--------------------------------------------------------------------------------------------------------------------The sum of the measures of the interior angles for a polygon with n sides is (n  2)  180 .
The measure of each interior angle of a regular n-gon is
(n  2)  180
.
n
The sum of the measures of the exterior angles, one at each vertex, for any convex
polygon is 360 .
360
Therefore, the measure of each exterior angle for a regular polygon is
.
n
The students must also know the names for polygons. For example, a heptagon has seven sides.
Review of Points of Concurrency
C
1.
D
A
Name_________________________
Let point E be the centroid of triangle ABC. If BE = 42,
what is BD?
E
B
--------------------------------------------------------------------------------------------------------------------G
For Questions 2-3, consider the figure to the left. FH is an altitude of
FGI . Let FH = 16, and HI = 30.
H
F
2.
True or False: GH must equal 30.
3.
What is FI?
I
--------------------------------------------------------------------------------------------------------------------4.
Darren has constructed a triangle, and he wishes to construct a circle circumscribed
around the triangle. Which of the following segments/lines should he construct in order
to find the center of the circumscribed circle?
A)angle bisectors
C)medians
B)perpendicular bisectors
D)altitudes
--------------------------------------------------------------------------------------------------------------------In the figure to the left, LKJ is a right angle, and point M is the
circumcenter of LKJ .
L
K
5.
True or False: LM must be equal to MJ.
6.
True or False: A perpendicular bisector for this triangle must run
thru vertex K.
M
J
--------------------------------------------------------------------------------------------------------------------N
In the figure to the left, point Q is the incenter of NOP .
7.
True or False: NQ must be equal to QO.
8.
If the distance from point Q to side OP is 8 inches, what is the
distance from point Q to side PN .
Q
O
P
--------------------------------------------------------------------------------------------------------------------9.
What is the definition of concurrent lines?
T
10.
S
In the figure to the left, point R is the circumcenter of
triangle UTS. If RT  8x 1 , and SR  6x  6 , what
is the value of x?
U
R
--------------------------------------------------------------------------------------------------------------------W
For Questions 11-13, consider the figure to the left. Point Z is
A
the centroid of WVX , and AZ  5 y  2 , and AX  99 .
Z
V
X
11.
What is the value of y?
12.
What is XZ?
13.
True or False: WZ must be equal to VZ.
--------------------------------------------------------------------------------------------------------------------E
For Questions 14-16, consider the figure to the left. FDG is
a right triangle. Give the letter that corresponds to the
D
requested point.
C
F
B
14.
circumcenter
15.
orthocenter
16.
incenter
G
A
--------------------------------------------------------------------------------------------------------------------J
17.
l
O
K
In the figure to the left, line l is a perpendicular bisector of
side KM . Also, K is a right angle. If JO  13 ,
and KN  12 , what is ON ?
M
N
--------------------------------------------------------------------------------------------------------------------18.
Why is the point of concurrency for the angle bisectors of a triangle called the incenter?
--------------------------------------------------------------------------------------------------------------------T 19.
In the figure to the left, the distance from point O to ST is
O
9 units, and OT = 15 units. If OR is an altitude of OST ,
then what is RT?
R
S
Z
Y
Name ________________________
For Questions 20-22, consider the figure to the left. XJY is
an obtuse angle. Give the letter that corresponds to the
requested point for XJY .
C
M
X
20.
orthocenter
21.
circumcenter
22.
centroid
J
R
--------------------------------------------------------------------------------------------------------------------For Questions 23-25, give the vocabulary word that best describes the requested geometric
figure.
23. QS
24. TV
P
25. SW
T
O
W
O
T
V
S
Q
R
S
S
--------------------------------------------------------------------------------------------------------------------Other Important Information
Vocabulary terms/phrases: concurrent, perpendicular, acute triangle, right triangle, obtuse
triangle, altitude, orthocenter, median, centroid, perpendicular bisector, circumcenter, a circle
circumscribed around a polygon, angle bisector, incenter, distance from a point to a line
Important Properties: all three properties concerning a median, the property concerning the
incenter, the property concerning the circumcenter
Sketches: you should be able to sketch an altitude, the three altitudes for a right triangle or an
obtuse triangle, the medians, the perpendicular bisectors, the perpendicular bisectors for a right
triangle or an obtuse triangle, the angle bisectors
Really Important: you should know the location of each of the points of concurrency for an
acute triangle, a right triangle, and an obtuse triangle
Other Stuff: the origin of the terms incenter and circumcenter, the ability to recognize one of the
vocabulary terms/phrases when given a picture
Review of Triangle Congruency
1.
Name the six theorems by which one can prove that two triangles are congruent.
2.
As instructed in class, name the three ways that do not prove triangle congruency.
3.
In order to use the HL Theorem to prove triangle congruency, what three things must one
prove first?
4.
Fill in the blank: In order for alternate interior angles to exist, two lines (or segments)
must be _______________, and they must be intersected by a transversal.
5.
Name the three special quadrilaterals that have congruent diagonals.
6.
Name the three special quadrilaterals that have perpendicular diagonals.
7.
Name the four special quadrilaterals in which the both diagonals bisect each other.
8.
Name the four special quadrilaterals in which both pairs of opposite sides are parallel.
9.
A triangle has two sides which measure 5 feet and 6 feet, and an interior angle that
measures 65 . Another triangle also has two sides that measure 5 feet and 6 feet,
and an interior angle that measures 65 . Is there enough information to prove these
two triangles are congruent? Explain why or why not.
--------------------------------------------------------------------------------------------------------------------For Questions 10-15, if there is not enough information given or marked to prove that the two
triangles are congruent, write "not enough information". If there is enough information, state the
theorem by which it would be best to prove triangle congruency.
10.
13.
16.
11.
14.
12.
15.
One right triangle has sides that measure 9 inches and 12 inches. Another right triangle
has sides that measure 9 inches and 12 inches. There is not yet enough information to
prove that these triangles are congruent. Draw a picture of a scenario that matches this
description in which the triangles would not be congruent.
Name ________________________
Write three unique segment congruency statements and three unique angle congruency
statements based on the following triangle congruency statement: SLY  WAR .
17.
--------------------------------------------------------------------------------------------------------------------For Questions 18-21, information has been provided through the markings on a figure or the
description. In addition, by examining the figure, you can determine certain congruencies
through properties and theorems like the Reflexive Property or the Vertical Angles Theorem.
However, what additional information would still need to be given in order to prove that the
triangles are congruent by the requested theorem? You must write a congruency statement.
BEC  DEC by HL
18.
19.
XCA  XHO by AAS
C
C
O
B
E
D
A
H
VIP  MRS by ASA
20.
X
21.
Let VP  MS , and V  S .
QKT  UZF by SSS
Let QK  UZ , and KT  ZF .
--------------------------------------------------------------------------------------------------------------------R
N
For Questions 22-25, consider parallelogram RNDZ to the left.
Provide the reason (as in a proof) why the statement is true.
E
Z
D
22.
RZ  ND
23.
REZ  NED
24.
NRD  ZDR
25.
RE  ED
26.
Given: OS is a median of JON , and JO  ON .
N
S
Prove: SOJ  SON .
J
Statements
O
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
--------------------------------------------------------------------------------------------------------------------V
T
27.
Given: TV is parallel to XY , and W is the midpoint of XV .
Prove: TVW  YXW by ASA Theorem
W
Y
X
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
--------------------------------------------------------------------------------------------------------------------B
28.
Given: AC bisects BAD , and AB  AD .
Prove: BAC  DAC by paragraph proof
A
C
D
Review Sheet for Statistics
Name_________________________
Terms/Ideas to Know
Be able to construct a statistically-based argument. In other words, be able to explain why one
data set is better than another using statistics.
What is a representative sample? When given a real-life scenario, what would be the most
appropriate way to obtain a representative sample?
What is a random sample? What are ways in which one can be obtained?
What are the measures of center? What are the measures of spread (variability)? Be able to
compare each of these measures for two different data sets in order to determine which is better
or more consistent.
Understand that a higher measure of spread indicates that the data is more "spread out", and,
hence, less consistent.
Be able to calculate the mean and the mode of a data set.
Be able to calculate the median for sets with an odd or even number of data values.
Be able to calculate the upper quartile, lower quartile, and interquartile range for sets with a
different number of data values.
Be able to calculate the mean absolute deviation of a set. Understand what the mean absolute
deviation represents.
Find the mode and median of a data set based on its histogram.
Find the median, interquartile range, outliers, and other information based on a box and whisker
plot. Also, be able to compare two box and whisker plots.
Be able to interpret a stem and leaf plot.
Be able to find outliers, and determine the "normal range of values".
--------------------------------------------------------------------------------------------------------------------Here is a chart from the quiz from earlier in this unit. It displays the number of touchdown
passes per year for quarterback Tom Brady. Use this information for Questions 1 & 2.
Year
TD's
2001
17
2002
28
2003
23
2004
28
2005
26
2006
24
2007
50
2009
28
1.
In 2007, Tom Brady set an NFL record by throwing 50 touchdown passes in one season.
Based on Brady's statistics, would 50 touchdown passes be considered an outlier?
2.
What would be largest number of touchdowns that he could throw in one season that
would NOT be considered an outlier (your answer must be a whole number)?
--------------------------------------------------------------------------------------------------------------------3.
A management team seeking to improve customer service wants to get a representative
sample of the customers at a Barnes and Noble bookstore. Which of the following would
provide the most representative sample?
A)Talk to the customers who are dining at the coffee shop inside of the store.
B)Set up a table next to the train set in the Kids' book section, and talk to those who pass.
C)Conduct an on-site survey involving every tenth person at the checkout line.
D)Place phone calls to every customer who owns a Barnes and Noble VISA card.
Suppose the two box and whisker plots below represent all of the test grades for Odell Bettis
(top) and Jeremiah Kingman (bottom) in one school year. Use the plots for Questions 4-5.
70
75
80
85
90
95
100
70
75
80
85
90
95
100
4.
Using a measure of center taught in class, explain which student is the better student.
Use complete sentences in your explanation.
5.
Using a measure of variability taught in class, explain which student is more
consistent. Use complete sentences in your explanation.
---------------------------------------------------------------------------------------------------------------------
The histograms above compare the number of wins per year for the New York Yankees to the
Boston Red Sox since 1985. For example, the Yankees won between 70 and 79 games five
times during that time span.
6.
State the mode number of wins for both teams. By comparing the modes, make a
statement about which team has been better over the last 25 years.
7.
State the median number of wins for both teams. By comparing the medians, make a
statement about which team has been better over the last 25 years.
Name ________________________
The U.S. Census Bureau tracked the lives of 10 men and 10 women born in the year 1900. It
recorded the ages (in years) at death for each of the participants. The two stem and leaf plots
below display these ages.
Men
Women
8.
Find the mean ages at death for both
men and women. Based on this information,
0
0
1
which gender should one expect to have lived
1
7
1
longer?
2
5
2
3
48
3
2
9.
Find the mean absolute deviations of both
4
33
4
36
5
8
5
2
data sets. Based on this information, for
6
02
6
which group is the mean a more accurate
7
5
7
666
predictor of life expectancy?
8
8
17
--------------------------------------------------------------------------10.
The following table shows the amount of calories per serving of certain types of
Kellogg's cereal.
Cereal
Apple Jacks
Cocoa Krispies Corn Pops Cookie Crunch Froot Loops
Calories 110
120
110
110
100
Cereal
Frosted Flakes Rice Krispies
Smacks
Calories 130
130
100
A)What is the median number of calories?
Raisin Bran
190
B)What is the interquartile range?
C)What is the normal range of values for this data set in which a value must fall
in order to NOT be considered an outlier?
D)Is Raisin Bran's 190 calories an outlier in this set?
--------------------------------------------------------------------------------------------------------------------11.
Suppose a teacher wants a random sample of his classroom. Why would "calling on
those who raise their hands" NOT provide a random sample?
--------------------------------------------------------------------------------------------------------------------*12. One data set has a median of 30 and an interquartile range of 6. A second data set has
a median of 25 and an interquartile range of 10. What is the lowest possible value for
a lower quartile that could appear simultaneously in both sets? What is the highest
possible value for a lower quartile that could appear simultaneously in both sets?
13.
The following list provides CO2 emissions levels from 1999 for the following countries:
Country
CO2 Level
County
CO2 Level
China
2.3
Brazil
1.8
India
1.1
Russia
9.8
United States
19.7
Pakistan
0.7
Indonesia
1.2
Bangladesh
0.2
A)What is the interquartile range of this data set?
B)Give a statistical reason why the United States' CO2 level is much worse than the rest
of the world's.
Review of Methods of Counting
1. American Idol is down to 10 females and 10 males. If only one can win the title and you can
only vote for one contestant, for how many options do you have to vote?
2. There are 8 girls on the Varsity girls' tennis team and 7 guys on Varsity boys' tennis team. If
one girl and one guy can each win most improved, how many different pairings of a girl and a
guy can win the award at the end of the season?
3. Susie is competing in the Miss LHS pageant. There are 24 girls competing but only the top 5
win a trophy (1st-5th). How many different ways are there for the top 5?
4. Matt’s family is going to the Columbia Zoo this weekend. They have time to see 5 exhibits,
and the order that they see the exhibits does not matter. If there are 9 exhibits total, to how many
combinations of exhibits can they go?
5. Jessie decided to start a scrapbook, and she is going to Hobby Lobby this weekend to buy the
scrapbook, pages, and stickers. She only has enough money to buy 1 scrapbook, 10 pages, and 5
sets of stickers. There are 10 scrapbooks, 50 pages, and 100 sets of stickers available in the store.
How many options does Jessie have?
6. Kayla is in charge of a talent show at BHS. There are eight contestants. She must choose in
what order they will appear. How many different ways can she schedule the performers?
7. A college graduate has job offers from 4 insurance companies with headquarters in Los
Angeles, 2 companies with headquarters in New York, and 3 companies with headquarters in
San Francisco. How many different choices of companies are available to her?
8. Lakeside High School faculty are issued special coded identification cards that consist of four
letters of the alphabet. How many different ID cards can be issued if the letters can be used more
than once?
9. Thirteen people were trying to be one of the first three callers to a radio station. How many
different sets of people could have succeeded?
Name ________________________
10. There is a new game in Las Vegas where you flip a coin and roll a dice simultaneously.
There are two sides to each coin and six sides to each die. To win the maximum prize, the coin
needs to land on heads and the dice needs to land on a number larger than 3. How many ways
can this happen?
11. Knowing it was wrong but scared because he didn’t study, Andy tried to cheat on his math
test. However, he was not very good, because when he tried to see the paper in front of him, all
he could see of the answer to #3 was that it was 5 numbers long, the first number was 7, 8 or 9,
and the last number was 5 or 8. How many different answers could Andy put for question #3?
12. Ms. Middlebrooks’ favorite place to shop is Ann Taylor Loft. Knowing this, her mom got
her a $100.00 gift card for her birthday. When Ms. Middlebrooks went to use it, she found that
she could afford only 3 shirts or 2 pair of pants, but she likes 6 of the shirts on display and 4
different pairs of pants. How many different ways could Ms. Middlebrooks’ spend her gift card?
13. Tom loves Wild Wing Cafe's buffalo wings, but he also really likes their wraps. If there are
7 meal options containing buffalo wings and 4 wrap meals, how many choices does he have for a
meal?
14. Mary LOVES ice cream, so her sweet boyfriend treated her to Cold Stone Creamery as part
of their date on Friday night. She knew she wanted the Love It cup (the big one) with three
scoops of her favorite flavors, but there is a BIG problem… she likes all fifteen flavors that they
offer. It was taking her so long that her boyfriend asked her, “Seriously Mary. There can’t be
that ways for you to fill your cup!” How many ways are there?
15. At the Winter Olympics, Apollo is racing for the gold in speed skating against the ten top
speed skaters in the world. Knowing that the top three finishers receive gold, silver, and bronze
medals respectively, how many different ways could the audience possibly see these skaters on
the medal stand?
Other Things to Know for The Test
Be able to explain the differences between each type of problem (addition principle of counting,
multiplication principle of counting, permutation, and combination).
Be able to demonstrate work with evaluating factorials.
Be able to write down formulas for permutations and combinations using n and r.
Probability Review
Name_________________________
1.
The United States has two runners in the 400m final at the Summer Olympics. If there
are twelve runners in total, what is the probability that the U.S. runners finish in 1st and
2nd place. For the purposes of this problem, assume all runners have an equal chance of
winning.
2.
During a game of Blackjack, there are 48 cards remaining in the deck. Poncho needs for
the next card dealt to him to be an Ace or a 2 in order to keep from busting. If all of the
Aces and 2's are still in the deck, what is the probability that the next card dealt to him is
a card he desires?
3.
Diane is playing Yahtzee, and she is going to roll two dice at the same time, and she
hopes that at least one of them lands on a 4. What is the probability that at least one of
the dice will land on a 4?
For Questions 4-8, consider the Wheel of Fortune to the right.
There are 24 spaces on the wheel.
4.
What is the probability of landing on Bankrupt or Lose
a Turn?
5.
What is the probability of landing on a $500 space?
6.
What is the probability of landing on a $5000 space on
the first spin, followed by a Bankrupt space on the
second spin?
7.
What is the probability of landing on a $300 space or a
$400 space in one spin?
8.
What is the probability of landing on a space greater
than $699 in three consecutive spins?
9.
An incoming freshman in the fall of 2009 has to take Math I. If Ms. Middlebrooks
teaches four classes, Mr. Middleton teaches four classes, and Ms. Whitmire teaches two
classes, and Ms. Steflik teaches two classes, what is the probability the student will get
Ms. Middlebrooks? Assume all classes are of equal size, and the student has an equal
chance to get any of the teachers.
10.
In Major League Baseball, there are five teams in the following divisions: AL Central,
AL East, NL East, and NL West. There are four teams in the AL West, and there are six
teams in the NL Central.
A)What is the probability that a team from the NL West or the AL West wins the
championship? Assume all teams have an equal chance of winning and that only one
team can win the championship.
B)What is the probability that a team from an "NL" division wins the championship?
Suppose there is a new casino game in Atlantic City involving two dice that costs $10 to
play. If a person rolls the same number of both dice, then he/she receives $60.
Otherwise, the person receives no money. What is the expected value of one playing of
this game?
11.
12.
What is the probability of drawing a Spade or a King in one draw from a standard deck?
13.
What is the probability of drawing three Diamonds in a row from a standard deck if the
cards are not replaced when drawn?
14.
What is the probability of drawing two 5's from a standard deck if it is already known
that the first draw is a 5? Assume that the first card is not replaced.
15.
There are 231 people in the audience at Live with Regis and Kelly. On a particular day,
the entire 5-person Rogers' family is in attendance. If one audience member will be
randomly selected to receive a trip, what is the probability that the person selected in a
member of the Rogers' family?
16.
What is the probability that the sum of two dice rolled simultaneously is an 11 or 12?
17.
What is the probability that the sum of two dice rolled simultaneously is a perfect
square number or an even number less than 7?
18.
There is a card game in which a person draws a card from a standard 52-card deck. If
he/she draws a numbered card, then the person gets 5 points. If he/she draws a face card,
then he/she gets 10 points. If the person draws an ace, then he/she gets 25 points. What
is the expected value of one draw from a full deck?
19.
You have four contestants that you really like on the TV show Survivor. However, there
are sixteen total contestants on the show.
A)On the first show, one of the contestants is eliminated. What is the probability that the
contestant eliminated is not one of your favorites? Assume all of the contestants have an
equal chance of being eliminated.
B)What is the probability that all four of your favorites finish 1st, 2nd, 3rd, and 4th?
20.
A whole number is randomly selected from 1 to 20.
A)What is the probability that the number is greater than 11?
B)What is the probability that the number is prime?
C)What is the probability that the number is 8 or 9?
D)What is the probability that the number is an odd number or a multiple of 3?
E)Suppose two different numbers are selected. What is the probability that they are both
less than 10?
F)Suppose two numbers are selected, and it is possible to select the same number twice.
What is the probability that both numbers are multiples of 5?
Review of Quadratic Equations
Rewrite each equation in standard form, and solve for x. If it cannot be solved by factoring,
state “Cannot be solved by factoring”.
1. x 2  x  12  0
2. 6 x 2  14 x
3. x 2  2 x  100  2 x
4. 14 x 2  8  12 x  13x 2  18 x
5. 11x  4 x 2  5 x 2  18
6. x 2  3  0
7. 4 x 2  10 x  10  3  3x  3x 2
8. 3  2x  x 2
9. x 2  10  11x
10. 4 x 2  12 x  4  2 x 2  2 x  4
11. x 2  74  10
12. 4 x 2  8 x
--------------------------------------------------------------------------------------------------------------------Predict the number of solutions to the quadratic equation based on the table or list of ordered
pairs.
13.
x
2
3
14.
y
-3
2
15.
x
1
2
3
4
5
y
8
5
4
5
8
16. {(2, 0), (1, 4), (0, 6), (1, 6), (2, 4), (3, 0)}
x
-1
0
1
2
y
-6
-4
-4
-6
17.
x
y
-3
1
-2
0
-1
1
0
4
1
9
Name ________________________
State the solutions to the quadratic equations based only on its graph.
18.
19.
4
4
2
2
-5
-5
5
-2
-2
-4
-4
-6
20.
21.
4
4
2
2
-5
5
-5
5
-2
-2
-4
-4
-6
-6
--------------------------------------------------------------------------------------------------------------------h  16t 2  d
in which h is the height in feet above the ground
t is the time in seconds since it was dropped
and d is the height at which the object was dropped
22.
A squirrel in a tree dropped an acorn 48 feet to the ground. How many seconds did it take
for the acorn to reach the ground? Round your answer to the nearest tenth of a second.
23.
The tourist at the Statue of Liberty drops their drink from the top of the statue at a height
of 256 feet. How long will it take for the drink to hit the ground?
24.
An art teacher painted a rectangular picture on the art room wall that was 7 ft x 9 ft.
Then she enlarged it by increasing both the width and the length by x feet. She only has
enough materials to paint 36 more square feet. What is the maximum width she can add
to her painting?
25.
The Turner’s have a rectangular backyard garden and want to expand it. They want to
add x amount of feet onto both the east side and south side of the garden. The current
dimensions of the garden are 4 ft. x 6 ft., and they have enough materials to add 11 more
square feet to the garden. If the new garden will also be rectangular, what is the
maximum width of the expansion to the garden?
26.
A right triangle has a leg that measures 12 cm and a hypotenuse that measures 15 cm.
What is the length of the other leg?
Review of Rational and Radical Equations
Solve for the variable. Express all answers as exact decimals or as fractions in simplest form.
1.
x  49
2.
20  y  6
3.
z  8  5
4.
a  15
5.
b  11  10
6.
164  4 c
7.
3 d  12  36
8.
5   k  6  14
10.
2 n  6  1  23
9.
4m  8  1  11
11.
4 p3  2
12.
41  4 5r  3
13.
12 9u  4  6  150
14.
5 4

x 11
15.
3 y 10

7
2
16.
10
 34
a 1
Name ________________________
17.
800 25b

b
2
18.
c  3 c  11

4
9
19.
d 2 3

33
d
20.
1
k 8

k 5
4
21.
20 m  4

m
m
22.
3 1
 8
n 4
23.
p 1 p  2

 1
5
4
24.
4
2 2

3s 7
25.
u  4 36

4
5
5u
26.
8
45 w

w 3
27.
A car travels 125 miles in 3 hours. How far would it travel in 5 hours?
28.
Mrs. Field's cookie recipe calls for 4 cups of sugar. There are 8 tablespoons in ½ cup of
sugar. How many tablespoons does Mrs. Fields need to bake the batch of cookies?
29.
Mrs. Belcher is creating the schedule for the LHS diving team practices. It takes each
diver 2 minutes to practice a dive because they must first adjust the board and then dive.
During practice the divers are at the diving well for 70 minutes. How many dives can
occur during the practice?
***You must also know how to check solutions to radical equations on a calculator.***
Review of Coordinate Geometry
Name________________________
1. The __________________________ of a segment is the same distance from both endpoints.
2. What is the distance between two points with coordinates of (1,-5) and (-5,7)?
3. What is the distance from point A to point C?
4. What is the distance from point B to point
D?
6
4
A
2
-5
B
5
-2
D
5. What is the distance from point A to point
D?
C
6. Find the midpoint of the segment with
endpoints at (-4, 6) and (3, -8).
-4
-6
7. Consider segment EF with a midpoint
named M. If the coordinates of point M
are at (-6.5, -0.5), and the coordinates of
point E are at (10, -5), what are the
coordinates of point F?
8. Find the midpoint of the segment with endpoints at (3,2) and (-5, 2).
9. A segment has one endpoint at (15, 22) and a midpoint at (5,18), what are the
coordinates of the other endpoint?
10. Joe begins walking in a straight line at point (-7, -4) and decides to stop and rest at
point (-2, 2). Joe is only halfway to his destination when he stops to rest. What are
the coordinates of Joe's destination?
11. The endpoints of a diameter of a circle are (-6, -2) and (2,4).
a. Find the length of the diameter.
b. Find the midpoint of the diameter.
c. Find the length of the radius (distance from one endpoint of the diameter to
the midpoint of the diameter).
Classify the triangle or quadrilateral plotted on the coordinate plane or with the given vertices.
Prove the classification.
12. Quadrilateral ABCD with vertices at
A(1,1), B(2,5), C(4,6), and D(8,5).
13. Triangle EFG graphed below
4
6
4
2
F
2
-5
-5
5
5
10
E
-2
-2
G
-4
-4
-6
14. Quadrilateral KLMN graphed below
X(-4,-3), Y(0,-4) and Z(3,-1).
15. Triangle XYZ with vertices at
6
6
M
4
4
2
L
2
N
-5
-5
5
5
-2
K
-2
-4
-4