Download VOCABULARY – Write definitions for the math terms below and

NSSHS
Geometry
Midwinter
Review
Packet
2016
Name: _____________________________
Due Date:
_____________________
This is your first grade.
1
Due Date:
_______________________
Attached is your midwinter review packet for Geometry.
This is your first grade.
You MUST SHOW WORK in order to receive credit. This
means if you typed something into a calculator to solve
it, you must write what you typed so I know how you found
the answers.
NO WORK = NO CREDIT (GRADED FOR CORRECTNESS)
The problems are on the work you’ve covered in Algebra 1.
Use your old notes,tests, or quizzes to help you, and if
possible, the internet.(kahn academy, purple math, etc. )
“I didn’t know how to do that one”
credit.
will not get you
Try something, even if it is wrong.
NOT HAVING A CALCULATOR IS NO EXCUSE
FOR NOT COMPLETING A PROBLEM.
FIND A WAY.
If you do not hand this packet in or on ______________,
OR if there is no work with your answers, then you will
receive a O as your first grade.
If you need more room, just attach any papers with work
on them with problem numbers labeled.
If you attempt to complete this packet the night before
it is due, you will most likely not finish, or not have
all the work I’m asking for. I suggest you do a little
at a time.
2
Objectives for Geometry Midwinter Packet
2016
I. Finding the Equation of a Line ( Problems: #1- 8)



Given a point that lies on that line and the y-intercept
Given a point and a parallel line
Graphing using the slope-intercept form
II. Solving Equations ( Problems: #9-22)






Solving equations with variables on both sides
Using order of operations
Using properties of equality
Solving inequalities
Solving literal equations
Solving absolute value equations
III. Systems of Equations ( Problems: #23-27)


Using the linear combination method to solve systems of equations
Using the substitution method to solve systems of equations
IV. Radicals ( Problems: #28-43)




Simplifying radicals
Squaring radicals
Rationalizing radicals
Exponents
V. Proportions ( Problems: #44-50)






Solving proportions by cross multiplying
Solving proportions using equivalent fractions
Solving equations involving inverse operations
Scale factor
Percent and tip.
Simplifying Ratios
VI. Factoring ( Problems: #51-57)



Solving quadratic equations by taking the square root of both sides
Using properties of equality
Multiplying binomials (FOIL)
3
VII. The Pythagorean Theorem ( Problems: #58 -62 )


Using the Pythagorean theorem to find missing lengths in right triangles
Using properties of equality
VIII. Polynomials ( Problems: #63 - 65)

IX.


Simplifying polynomials
Quadratic Equations (Problems: #66-70)
Solving quadratic equations by taking the square root of both sides
Using properties of equality
4
Finding the Equation of a Line
ALGEBRA REVIEW
Example: Find an equation of the line, in slope intercept form, that passes through
the point (3, 4) and has a y-intercept of 5.
y = mx + b
Write the slope-intercept form.
4 = 3m + 5
Substitute 5 for b, 3 for x, and 4 for y.
-1 = 3m
Subtract 5 from each side.
1
 =m
3
Divide each side by 3.
1
1
The slope is m   . The equation of the line is y   x  5 .
3
3
1
The slope of the parallel line is  .
3
Write the equation of the line, in slope intercept form, that passes through the given
point and has the given y-intercept.
1.
(2, 1); b = 5 _________________
3.
(-2, -1); b = -5 ________________
4.
2.
(7, 0); b = 13 ________________
Write the equation of a line that passes through (5, 1) and is parallel to
3
y  x  4 . (Hint: use the slope-intercept form to solve for b this time)
5
5
Graphing Linear equations
To graph a linear equation use the slope(m) and y intercept(b). First graph the b then
count the slope up and over or down and over (if negative slope) (remember it has to
be in slope-intercept form first- solve for y!!)
For example :
2
x  3 , you would plot a point at  3 on the y-axis then count up 2 and over to the
3
right 3 units to plot another point, and connect the dots to make the line.
y
5. Graph y 
1
x 1
2
6. Graph y  
7. Graph y = 2x − 1
2
x4
5
8. Graph 3x  4 y  4
6
Solving Equations with Variables on Both Sides
Examples:
a. 6a – 12 = 5a + 9
b.
a – 12 = 9 Subtract 5a from each side.
a = 21 Add 12 to each side.
6(x + 4) + 12 = 5(x + 3) + 7
6x + 24 + 12 = 5x + 15 + 7
6x + 36 = 5x + 22
x = -14
Solve the equation.
9.
3x + 5 = 2x + 11 _______________ 10.
y – 18 = 6y + 7 ______________
11.
-2t + 10 = -t ________________
12.
54c – 108 = 60c ____________
13.
x + 6 = 9_______________
2
1 + j = -14
7
15.
4x + 2(x – 3) = 0 _______________ 16.
14.
7
_____________
3 + m = -10 ________________
Solving Inequalities.
Examples:
Solve like an equation.
a. 3y + 1 > y − 3
2y + 1 > −3 Subtract y from both sides
2y > −4 Subtract 1 from both sides
y > −2 Divide by 2
b. 4x + 3 < 8x + 15
−4x + 3 > 15 subtract 8x from both sides
−4x > 12 subtract 3 from both sides
x < −3 divide by −4 (need to flip symbol)
17. 3x + 2 < 2x + 5
18. 4 − 5y  8 − y
17. _______________
18. _______________
Solving Literal Equations
Example:
C = 2  r Solve for r
C
r
Your goal is to isolate the variable by inverse operations.
2
19. Volume of a rectangular prism is V = lwh . Solve for h .
20. Area of a square is A =  r . Solve for r.
19. _________________
2
20. ___________________
Solving absolute value equations.
Examples:
x  3 10
x  7 subtract 3 from both sides
x  7 or x   7 since the absolute value of 7 or −7 is 7.
21. x  8 10
22. x  12  35
8
21. __________________
Solve the System of Equations:
22. _________________
Example 1: Linear Combination Method
4 x – 3 y = -5
-4 x + 2 y = -16
The goal is to obtain coefficients that are opposites for one of the variables.
4 x – 3 y = -5
-4 x + 2 y = -16
-1 y = -21
-1
-1
y = 21
Substitute 21 for y: 4(21) – 3 y = -5. Solve to get y = -1. The solution is (21 , 89/3)
****************************************************************************
Example 2: Substitution Method
3 x + 2 y = 16
x + 3 y = 10
x = 10 – 3 y
Now substitute 10 – 3 y for x in the first equation: 3(10 – 3 y) + 2y = 16.
Solve for y to get y = 2.
Substitute 2 for y: x = 10 – 3(2). Solve to get x = 4. The solution is (4, 2).
****************************************************************************
23. 2 x – 3 y = -16
24. y = x - 2
25. y = 4x - 8
y=5x+1
2x+2y=4
y = 2x + 10
26. y = x + 1
Y = 2x - 1
27. 7 x + 2y = 10
-7 x + y = -16
9
Simplifying Radicals
An expression under a radical sign is in simplest radical form when:
1) there is no integer under the radical sign with a perfect square factor,
2) there are no fractions under the radical sign,
3) there are no radicals in the denominator
Express the following in simplest radical form.
28.
33.
50
29.
24
30.
13
49
34.
192
31.
169
6
32. 147
3
35.
27
6
IV. Properties of Exponents – Complete the example problems.
PROPERTY
Product of Powers
Power of a Power
Power of a Product
Negative Power
Zero Power
Quotient of Powers
Power of Quotient
EXAMPLE
am · an = am + n
(am)n = am · n
(ab)m = ambm
1
a-n = n
(a ¹ 0)
a
a0 = 1
(a ¹ 0)
m
a
= am – n (a ¹ 0)
n
a
m
m
æaö = a
(b ¹ 0)
ç ÷
bm
èbø
x4 · x2 =
(x4)2 =
(2x)3 =
x-3 =
40 =
x3
=
x2
3
æxö
ç ÷ =
çy÷
è ø
Simplify each expression. Answers should be written using positive exponents.
36. g5 · g11 __________
38. w-7
37. (b6)3 __________
y 12
39.
__________
y8
__________
40. (3x7)(-5x-3) __________
42.
-15x 7y -2
25x -9y 5
41. (-4a-5b0c)2 __________
æ 4x 9
43. çç
4
è 12x
__________
10
ö
÷÷
ø
3
__________
Solving Proportions
x 3

Cross Multiply
8 4
4x = 8 • 3
4x = 24
x=6
Solve for the variable.
Examples:
a.
b.
6
1

Cross Multiply
x4 9
6•9=x+4
54 = x + 4
50 = x
44.
x 1
 _______________________ 45.
20 5
6 m

___________________
19 95
46.
3w  6 3
 ___________________
28
4
3
1
________________

p6 p
47.
Scale Factor:
Example: A map has a scale factor of 1in:15 mi, How far apart are two cities that are 5
inches apart?
1in
5in

Use a proportion
. Use cross products 1x = 15(5), x = 75 miles.
15mi x mi
48. A map has a scale factor of 4in:23 mi, How far apart are two cities that are 6
inches apart?
11
% and tip
Example: You and your friend’s go out to dinner. The bill total is $45.24. You want to
tip 18%. What is the total amount you should leave?
First find 18% of 45.24, by multiplying .18 times 45.24 = $8.15, then add that amount
to the $45.24. So, $45.24 + 8.15 = $53.39.
49. You and your friend’s go out to dinner. The bill total is $95.28. You want to tip
20%. What is the total amount you should leave?
Simplifying Ratios
Example: What is the ratio of 25x3 and 75x
25 x3
x2
write as a ratio first, then reduce
75 x
3
51. What is the ratio of 28x 3 y5 and 44x2y
****************************************************************************
Factoring and multiplying binomials.
Examples: a. x2 – 5x - 14
(x – 7) (x + 2)
x – 7 = 0 or x + 2 = 0
x = 7 or x = -2
b. (x + 3) (x - 5)
= x2 − 5x + 3x − 15 (FOIL)
= x2 − 2x − 15 (Simplify)
Factor each polynomial:
52.
x2 + 3x + 2 _________________
53.
x2 – x - 20 _______________
54.
x2 + 5x - 24 ________________
55.
x2 + 7x + 12 _____________
56.
(x + 8)(x − 9) = _______________
57. (2x − 3)(x − 4) = ________________
****************************************************************************
12
Pythagorean Theorem:
Examples:
c
a
b
a.
a = 12, b = 35, c = __________ b.
a = 10, b = _______, c = 26
a2 + b2 = c2
(12)2 + (35)2 = c2
144 + 1225 = c2
1369 = c2
1369 = c
37 = c
a2 + b2 = c2
(10)2 + b2 = (26)2
100 + b2 = 676
b2 = 576
b = 576
b = 24
c. A man leans a 8 ft. ladder against a house. The base of the
ladder is 2ft from the house. To the nearest tenth how high
on the house does the ladder reach?
8ft
2ft
Use Pythagorean’s theorem
x2 + 2 2 = 8 2
x2 + 4 = 64
x2 = 60
x  7.7 ft
Use the triangle above. Find the length of the missing side. Round answers to the
nearest tenth.
58.
a = 36, b = 15, c = _____________
59.
a = 17, b = _________, c = 49
60.
a = ________, b = 13, c = 24
61.
a = 19, b = 45, c = _________
62. A man leans a 12 ft. ladder against a house. The base of the
ladder is 4 ft. from the house. To the nearest tenth how high on the house does
the ladder reach?
13
Polynomials
Example:
Simplify the polynomial.
13x 2 + 2x – 4x + 15 – 3x -9
13x 2 - 5x + 6
combine the x values
combine the constants
8y 2 + 12x – 3y + 9 – 3x -9y
8y 2 + 9x – 12y + 9
combine similar terms
Simplify each expression.
63. y – 4 x + 16 y 2 - 10 x – 4 y 2
________________________
64.
________________________
5 m + 3 n – 4mn – 10 m – 8 n
65. 12 c 2 + 5 b – 8 c + 3 b 2 + 6 b – 8 c 2 ________________________
14
****************************************************************************
Solving Quadratic Equations
Example:
x2 – 5 = 16
x2 = 21
x =  21
Add 5 to both sides
Exercises: Solve.
66.
x2 + 3 = 13 ____________________67.
7x2 = 252 __________________
68.
4x2 + 5 = 45 __________________ 69.
11x2 + 4 = 48 ________________
70.
x2 -9 = 7 ________________
15