Download Chapter 1 Test

2nd Semester Geometry Final Exam Review 2015
Chapters: 5, 6, 7, 8, 10, 11, 12
Saturday, May 6, 17
-------------------------------------------------------------------------------------------------Vocabulary
Chapter 6
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Quadrilateral – a polygon with 4 sides
Parallelogram – a quadrilateral with both pairs of opposite sides parallel.
Rectangle – a parallelogram with 4 right angles
Square – A rhombus with 4 right angles
Rhombus – A parallelogram with all sides equal.
Trapezoid – a quadrilateral with 2 opposite sides parallel.
Diagonal –a segment that connects any 2 nonconsecutive vertices in a polygon.
Exterior angles
Interior angles
Midsegment of a trapezoid – A segment that connects the two midpoints of the legs of a trapezoid.
Chapter 7
1. Ratio
2. Scale factor
3. Proportion
4. Extended ratio
5. Similar Polygons and Triangles: AA, SAS, SSS similarity theorems
6. Triangle proportionality theorem
7. Similarity Statement
8. Proportional sides
9. Midsegment of a triangle
10. Special segments: angle bisector, altitude, median, pg. 501.
Chapter 8
1.
2.
3.
4.
5.
6.
7.
Geometric Mean
Pythagorean Theorem and its converse
Pythagorean Inequality Theorem
Special right Triangles: 45-45-90, 30-60-90, SOH-CAH-TOA
Trigonometric Ratios: Sine, Cosine, Tangent
Angle of Elevation and Angle of Depression, pg 580
Law of Sines, Law of cossines
Chapter 10
1
1.
2.
3.
4.
5.
6.
Circle, circle center, pg. 697
Radius, diameter, chord
Inscribed, circumscribed
Circumference of a circle
Minor arc, Major arc, semicircle, arc length
Central Angle, inscribed angle
2
Chapter 6 Review & Study Guide for chapter 6 test.
3
Ch6-1 Angles of Polygons, page 393
1. What is the sum of the measures of the exterior angles of a convex octagon?
The sum of the measures of the exterior angles of any convex polygon is equal to 360 degrees.
2. What is the measure of one exterior angle of a regular hexagon?
The Interior Angle Sum Theorem states that if a convex polygon has n sides and S is the sum of
the measures of its interior angles, then S = 180(n – 2). You can also use this equation to find each
interior angle by diving S by n, the number of sides.
3. What is the sum of the measures of the interior angles of a decagon (10-gon)?
4. The sum of the measures of the interior angles of a polygon is 720º.
How many sides does the polygon have?
Since S = 720, then 720 = (n – 2)180 and solve for n.
5. Find the value of x.
x = _____________
(Hint. what is the sum of the interior angles?)
6. Find the value of x.
7. Find the value of x.
8. Find the value of x and y,
x = ___________
If the figure is a parallelogram
y = ___________
x = _____________
(Hint. see question 1 above)
x = _____________
4
==============================================================
Ch6-2 Parallelograms, page 403
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
They also have:
 Opposite sides @
 Opposite angles are @
 Consecutive angles are supplementary
 Diagonals bisect each other
 Diagonals also divide the parallelogram into 2 @ triangles.
 If the parallelogram as one right angle, then it has 4 total right angles.
9. Find the value of x and y.
x = _________
10. Find the value of h and k. h = ________
If the figure is a parallelogram.
y = _________
If the figure is a parallelogram. k = ________
Complete each statement about parallelogram RSTU
1)
1) mSRU =
R
S
V
2)
2) SV =
U
T
5
==========================================================
Ch6-3 Tests for Parallelograms, 413
To determine if a quadrilateral is a parallelogram, you can do the following tests:
1. If both pairs of opposite sides of a quadrilateral are @
2. If both pairs of opposite angles are @
3. If the diagonals bisect each other
4. If one pair of opposite sides is both parallel and @ .
A quadrilateral only needs to pass one of the above tests to be proven a parallelogram
1. If both pairs of opposite sides of a quadrilateral are @ (use distance formula twice)
2. If both pairs of opposite angles are @
3. If the diagonals bisect each other. (Use the distance formula for each diagonal)
4. If one pair of opposite sides is both parallel and @ . (Use distance formula to prove one pair of
sides @ , and the slope formula to prove the same pair are ).
11. Determine whether the figure with vertices A(-5, 3), B(-1, 5), C(6,1), and D(2, -1)
is a parallelogram. Use the Distance and/or Slope formula appropriately.
Use graph paper to first plot the points and connect them to create the quadrilateral, then use the
formula(s) indicated on each of the tests above.
Ch6-4 Rectangles, page 423
The rectangle has all the properties of a parallelogram plus the following:
 The diagonals are @
 Has 4 right ’s
12. Find the value of x and y.
x = _________
y = _________
If the figure is a rectangle.
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===========================================================
Ch6-5 Rombi and Squares, page 430
The rhombus and the square have all the properties of a parallelogram.
The rhombus in addition has properties:
1) diagonals are ^ ,
2) all sides have equal lengths,
3) diagonals bisect the angles.
The square has all the properties of a rhombus plus all the properties of a rectangle.
13. Find the value of x, y and AB.
y = ________
x = ________
AB = ________
If the figure is a rhombus.
==========================================================
Ch6-6 Trapezoids and Kites, page 439
What is a trapezoid? A quadrilateral with exactly one pair of parallel sides. It is made up of 2 bases,
2 legs, and 2 sets of base angles.
What is an isosceles trapezoid? It is a trapezoid whose legs and diagonals are @ . Also each pair of
base angles are @ .
14. Find the measures of L, J, M.
L = _______
J = _______
M = _______
(Hint. The indicated marks show that the figure is an isosceles trapezoid.
7
What is a median? A segment that connects the midpoints of the legs of any trapezoid
The sum of the measures of the bases is twice the length of the median, so
b1 + b2 = 2(length of midsegment)
13. Find the value of x.
x = _________
What is a Kite? A kite is a quadrilateral with exactly two pairs of consecutive congruent sides.
Also, its diagonals are ^ and it has exactly one pair of opposite ’s @ .
14. Find the measures of V and T.
V = _________
The figure below is kite, so mV = mS.
T = _________
True or False
13. Diagonals are congruent in a rectangle and a square.
13. ___________
14. All angles are congruent in a square and a kite.
14. ___________
15. All sides are congruent in a parallelogram.
15. ___________
16. Base angles are congruent in an isosceles trapezoid.
16. ___________
17. All angles and sides are congruent in a rhombus.
17. ___________
18. Diagonals bisect each other in a parallelogram.
For more problems and vocabulary see pages 449 to 452.
18. ___________
8
Chapter 7 Review
A proportion is an equation that states that 2 ratios are equal.
Solve the proportion.
1.
3
5
=
m -1 3m - 4
m = __________
2.
Find the geometric mean of the two numbers.
3. 12 and 6
_____________
12 n
=
n 3
n = __________
Copy and complete the statement.
x
7
7 ?
y
4. If 9 = , then x = ?
_________________
5. ∆PQR is similar to ∆LMN.
a) Find the scale factor of ∆PQR to ∆LMN ________________
b) Find the value of x.
x = _____________
c) Find the value of y.
y = ____________
d) Find the value of z
z = ____________
===============================================================================
6. Find the length of RQ. _________________
(hint, triangle proportionality theorem, pg. 490).
===============================================================================
7. Find the value of y.
y = _________________
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8. Are the triangles to the right similar? _________
If yes, a) Explain why _________________________________
and b) Write a similarity statement ___________________
====================================================================================
9. Are the triangles to the right similar? _________
If yes, a) Explain why _________________________________
and b) Write a similarity statement ___________________
================================================================================
10. Find the value of a.
a = ________________
(Hint, triangle bisector theorem, pg. 504)
=================================================================================
11. A rectangular swimming pool has dimensions of 30 ft wide and 50 ft long. The rectangular
fence around the pool is similar to the shape of the pool. The fence is 45 ft wide.
Find the length of the fence.
Length =
__________________
12. Find the value of x.
A
x = ________________
12
B
4
C
x
18
E
D
10
Chapter 8 Review
===================================================================
Section 8-1, Pg. 538: If an altitude is drawn from one vertex of a right angle of a right triangle to
its hypotenuse, then the 2 triangles formed are similar to the given triangle and to each other. The
measure of this altitude is the geometric mean between the measures of the two segments of the
hypotenuse.
Moreover, the measure of a leg of the triangle is the geometric mean between the measures of the
hypotenuse and the segment of the hypotenuse adjacent to that leg. Practice the examples below
and apply these theorems.
8. Identify the three similar right triangles in the diagram to the right.
∆________ ∆________ ∆________
9. Find the length of x.
x = __________
10. Find the length of x. x = __________
----------------------------------------------------------------------------------------------------------------------------Section 8-2, pg. 547
2
2
2
Pythagorean theorem: c = a + b , where a and b are the measures of the length of the 2 legs of
the right triangle and c is the measure of the hypotenuse.
1. What is the longest side of a right triangle called? _____________________
2. What is the length of the hypotenuse of a right triangle with leg lengths of 3 in. and 4 in.?
A. 5 inches
B. 9 inches
C. 16 inches
D. 25 inches
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3. Find the length of r.
r = _________
4. Find the length of h. h = _________
5. Find the height and the area of the isosceles triangle below.
h = ______________
Area = ___________
----------------------------------------------------------------------------------------------------------------------Using the converse of the Pythagorean theorem, pg. 547
If the square of the longest side is equal to the sum of the squares of the 2 other legs of the
triangle, then the triangle is a right triangle.
6. Is ∆PYT a right triangle?
(yes/no) = _________
7. Classify the triangle formed by the side lengths 14, 21, and 25 as acute, right, or obtuse.
See page 550.
The geometric mean between 2 numbers is the square root of their product.
x = ab , where a and b are the 2 numbers and x is the geometric mean.
Find the geometric mean of the following pairs of numbers.
a. 4, 9
b. 16, 5
c. 12, 16
12
===================================================================================
Ch8-3 Special Right Triangles, pg. 558
A 45-45-90 triangle is the only isosceles right triangle. The hypotenuse is
leg, so the ratio of the sides is 1:1: 2 .
2 times as long as a
A 30-60-90 triangle also has special properties. The measures of the sides are x, x 3 , and 2x,
giving the sides a ration of 1: 3 :2. If you see these special triangles, you do not have to use the
Pythagorean theorem to find the sides of the right triangle.
Find the value of each variable.
1.
2.
3.
Find tan A and tan B. Write each answer as a fraction.
You might want to use, SOH CAH TOA, to remember the trigonometric ratios of the sides of a
triangle given an angle. These ratios help you find angles and sides of a right triangle.
6.
7.
Use a tangent ratio to find the value of x. Round your answer to the nearest tenth.
8.
9.
13
Use a sine or cosine ratio to find the value of each variable. Round to nearest tenth.
10.
11.
Solving a right triangle by finding all the angles and sides of the triangle given. Round your
answer to the nearest tenth. Remember that to find the angles, you must use the trigonometric
inverses. When you use your graphing calculator, use the second button before you press the cos,
sin, or tan.
12.
13.
14. A 30-foot tree casts a 12-foot shadow. Find the angle of elevation from the end of the shadow
on the ground to the top of the tree (to the nearest tenth of a degree).
Angle of Elevation = ____________
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Chapter 8 Review (Continued)
===================================================================
1. Which one of the following steps is not correct to find the length of x?
1. ____________
2. Find the height of the isosceles triangle below.
h = ______________
========================================================================
3. Determine whether the sides of each triangle given below form a right triangle. Show work.
a) 11, 20, and 21
___________
b) 12, 16, and 20
_____________
4. If the segment with length 12 in the figure below is the altitude and geometric mean of 16 and
x, then find the length of side x.
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5. According the Geometric Mean (leg) Theorem (8.3), the side x in the figure below is the
geometric mean of the side 2 and _______________. Finish the problem below to find the length
of x.
x = 2(__)
------------------------------------------------------------------------------------------------------------------------------Use special triangles definitions and trigonometric ratios to find x and y for the problems below.
Show that you can get the same answer using both methods.
6. Find side x and y.
side x = __________
side y = __________
7. Find sides x and y.
side x = __________
side y = __________
y
For problems 8-10 below. Use trigonometric ratios to find the length of the unknown sides.
Round your answer to the nearest tenth.
8.
8) side x = _______________
9.
9) side x = _______________
16
10.
10a) side x = _______________
10b) side y = _______________
===================================================================================
For problems 11-12 below, use a calculator to approximate the measure of P to the nearest
degree.
11. mP = _______________
12. mP = _______________
13. Solve the right triangle by finding the values of side x, mA, and m C. Round your answers
to angles to the nearest degree. (Each blank is 5 points – 15 points total)
side x = _______________
x
mA = _______________
mC = _______________
=====================================================================================
14. A 30-foot tree casts a 12-foot shadow. Find the angle of elevation from the end of the shadow
on the ground to the top of the tree (to the nearest tenth of a degree).
Angle of Elevation = _____________
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15. Find the geometric mean of the following pairs of numbers.
(Simplify completely and no decimals)
(3 pts each)
1
1
and
6
a) 9 and 16
b) 20 and 80
c) 2
15a. _____________
b. _____________
c. _____________
======================================================================================
16. Fill in the blanks:
(1 pt each blank)
a) The angle of __________________ is the angle formed by a horizontal line and the line of
sight to an object above.
b) The angle of __________________ is the angle formed by a ________________ line and
the line of sight to an object below.
c) The three trigonometric ratios on right triangles are S_____________, C_____________,
and T_______________.
d) You use the law of _____________ to solve a triangle if you know the measures of two
angles and any side (AAS or ASA).
e) You use the law of ______________ to solve a triangle if you know the measures of two
sides and the included angle (SAS)
=====================================================================================
17. Jon and David are playing basketball. Jon passes the ball to David when he is 24 feet from
him and 26 feet from the Goal. How far is David from the goal, if the angle from the goal to Jon
and then to David is 34 . Also state why you cannot use trigonometric
ratios in
David
d
this problem.
X = _________________
X
24 ft
34
Goal
Jon
26 ft
18
=====================================================================================
Challenge problem: Prove either the Law of Cosines or the Law of Sines (only one). 5 points.
a 2 = b 2 + c 2 - 2bcCosA
b 2 = a 2 + c 2 - 2acCosB
Law of Cosines: c = b + a - 2baCosC
2
2
2
sin A sin B SinC
=
=
b
c
Law of Sines a
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Chapter 10 Review
===================================================================
Use the diagram to match the notation with the term that best describes it.
1. ___________
2. ___________
3. ___________
4. ___________
5. ___________
6. ___________
7. ___________
---------------------------------------------------------------------------------------------------------------------------8. Find the value of x.
_________
9. Find the value of r. _________
(Point B and D are points of tangency)
10. AC and BE are diameters of circle F. Find the measure of the arcs:
a.
= __________
c.
= __________
b.
= __________
d.
= __________
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11. Find the value of x.
_________
12. Find the value of x.
_________
13. The measure of arc AC = 150 º
Find the measure of arc AB ________
14. Find the measure of
15. Find the measure of LNM _________
16. Find the measure of x and y.
Find the measure of arc LN __________
________
x = ______
y = ______
17. Line t is tangent to the circle.
Find the measure of
19. Find the value of x.
___________
_________
21
Chapter 11 Review
===================================================================
For each problem, write the area formula and show all of your work.
Make sure to label your answer with the appropriate unit of measure.
1. Find the area.
______________
2. Find the area.
______________
------------------------------------------------------------------------------------------3. Find the area.
______________
4. Find the area.
______________
5. Find the area.
______________
6. Find the area.
______________
---------------------------------------------------------------------------------------------------------------------------7. Area = 212 ft2
x = ___________
8. Area = 180 ft2
x = ___________
x
16
ft.
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9. The two polygons are similar.
9a. Write the scale factor ________
9b. Write the ratio of the perimeters ________
9c. Write the ratio of the areas ____________
9d. Find the area of the smaller polygon ____________
------------------------------------------------------------------------------------------------------------------------10. The ratio of the areas of two trapezoids is 121:36.
Write the ratio of the lengths of the corresponding sides.
___________
------------------------------------------------------------------------------------------------------------------------11. Find the circumference of a circle with radius 6 inches. __________________
12. Find the area of a circle with radius 6.5 ft.
__________________
13. Find the diameter of a circle with C = 64 ft.
__________________
14. Find the arc length of JK. __________
15. Find the length of radius AC. __________
-------------------------------------------------------------------------------------------------------------------------16. Find area of the shaded sector. _________
17. Find area of the regular polygon. _________
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Chapter 12 Review
===================================================================
Be sure to write the formula for each problem and show all of your work.
Make sure to label your answer with the appropriate unit of measure.
1. Find the number of faces, vertices, and edges of the polyhedron below.
Faces ________ , Vertices ________, Edges ________
----------------------------------------------------------------------------------------------------------2. Find the surface area and Volume.
Surface Area ____________
Volume _________
----------------------------------------------------------------------------------------------------------3 . Find the surface area. ____________
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--------------------------------------------------------------------------------------------------------------4. Find the surface area and Volume.
Surface Area ___________
Volume __________
-------------------------------------------------------------------------------------------------------------
5. Find the Surface Area and volume.
Surface Area ____________
Volume _____________
------------------------------------------------------------------------------------------------------------6. Surface Area = 717 in2 . Find x. __________
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