Name: ________________________ Class: ___________________ Date: __________

ID: A

Chapter 5 Practice Test Short Answer 1. Find the value of x. The diagram is not to scale. 40

x

40 32

25

25

2. B is the midpoint of AC, D is the midpoint of CE, and AE = 21. Find BD. The diagram is not to scale. C

B

D

A

E

3. Find the value of x.

16

3x – 4

1

Name: ________________________

ID: A

4. Find the length of the midsegment. The diagram is not to scale. 47

56

4x + 2

56

47

4x + 44

5. Q is equidistant from the sides of ∠TSR. Find the value of x. The diagram is not to scale.

T

|

| |

Q

|

)° 24

x+ (2 30°

R

S

6. DF bisects ∠EDG. Find the value of x. The diagram is not to scale. E

| |

8x + 42 F

)

15x )

D

30° G

2

Name: ________________________

ID: A

7. Find the center of the circle that you can circumscribe about the triangle. y 5 (–3, 3)

–5

5

x

(1, –2)

(–3, –2)

–5

8. Find the center of the circle that you can circumscribe about ∆EFG with E(4, 4), F(4, 2), and G(8, 2). 9. In ∆ABC, G is the centroid and BE = 9. Find BG and GE. C

B

G

A

D

E

F

10. Name a median for ∆ABC.

|

A

E

)

|

D

)

C

F B

3

Name: ________________________

ID: A

11. Name the point of concurrency of the angle bisectors.

12. For a triangle, list the respective names of the points of concurrency of • perpendicular bisectors of the sides • bisectors of the angles • medians • lines containing the altitudes. 13. What is the negation of this statement? Miguel’s team won the game. 14. What is the inverse of this statement? If he speaks Arabic, he can act as the interpreter. 15. Write the conditional statement illustrated by this Venn diagram.

Mammals

Cows

4

Name: ________________________

ID: A

16. Write the contrapositive of the conditional statement illustrated by this Venn diagram.

Dogs

Poodles

17. List the sides in order from shortest to longest. The diagram is not to scale. J 66° 50°

K

64° L

18. Two sides of a triangle have lengths 10 and 15. What must be true about the length of the third side? 19. m∠A = 9x − 7, m∠B = 7x − 9, and m∠C = 28 − 2x. List the sides of ∆ABC in order from shortest to longest.

5

Name: ________________________

ID: A

20. Identify parallel segments in the diagram. C

8 6 B

D 8 6

A

5

F

E

5

21. B is the midpoint of AC and D is the midpoint of CE. Solve for x, given BD = 5x + 3 and AE = 4x + 18. C

B

A

D

E

22. Write the inverse of this statement: If a number is divisible by two, then it is even. 23. To prove “p is equal to q” using an indirect proof, what would your starting assumption be? 24. Complete the indirect proof. Given: Bobby and Kina together hit at least 30 home runs. Bobby hit 18 home runs. Prove: Kina hit at least 12 home runs. Assume Kina hit a.___ than 12 home runs. This means Bobby and Kina combined to hit at most b.____ home runs. This contradicts the given information that c. _____. The assumption is false. Therefore, Kina d. ______. 6

Name: ________________________

ID: A

25. Complete the indirect proof. Given: Rectangle JKLM has an area of 36 square centimeters. Side JK is at least 4 centimeters long. Prove: KL ≤ 9 centimeters Assume that a. ____. Then the area of rectangle JKLM is greater than b. _____ , which contradicts the given information that c. _____. So the assumption must be false. Therefore, d. _____. 26. Li went for a mountain-bike ride in a relatively flat wooded area. She rode for 6 km in one direction and then turned and pedaled 16 km in another. Finally she turned in the direction of her starting point and rode 8 km. When she stopped, was it possible that Li was back at her starting point? Explain. Essay 27. If AC = 18 and BD = 21, find the perimeter of the small quadrilateral inside quadrilateral ABCD. Explain. Based on your work, make a conjecture about the relationship between the “midsegment quadrilateral” and the diagonals of the large quadrilateral. B

|

|

|

| C

||

||

||

||

A

D

28. AC and BD are perpendicular bisectors of each other. Find BC, AE, DB, and DC. Justify your answers. A 13 D

12

E

B

5 C

Other 29. Use indirect reasoning to explain why a quadrilateral can have no more than three obtuse angles. 7

ID: A

Chapter 5 Practice Test Answer Section SHORT ANSWER 1. 2. 3. 4. 5. 6.

64 10.5 4 42 3 6

1 7. (−1, ) 2 8. (6, 3) 9. BG = 3, GE = 6 10. BD 11. C 12. circumcenter incenter centroid orthocenter 13. Miguel’s team did not win the game. 14. If he does not speak Arabic, he can’t act as the interpreter. 15. If an animal is a cow, then it is a mammal. 16. If an animal is not a dog, then it is not a poodle. 17. LJ , JK , LK 18. less than 25

AB; AC; BC BD Ä AE, DF Ä AC, BF Ä CE, x=2 If a number is not divisible by two¸ then it is not even. p is not equal to q. a. fewer b. 29 c. Bobby and Kina together hit at least 30 home runs d. hit at least 12 home runs 25. a. KL > 9 centimeters b. 36 square centimeters c. The area is equal to 36 square centimeters d. KL ≤ 9 centimeters 26. No; for three segments to complete the sides of a triangle, the sum of the lengths of two segments must be greater than the length of the third segment.

19. 20. 21. 22. 23. 24.

1

ID: A

ESSAY 27. [4]

For the small quadrilateral, the top and bottom sides are both

1 2

AC , or 9, by the

[3] [2] [1]

1

BD, or 10.5 by the 2 Triangle Midsegment Theorem. Thus, the perimeter = 9 + 10.5 + 9 + 10.5 = 39. Conjecture: The sum of the lengths of the diagonals of a given quadrilateral is equal to the perimeter of the “midsegment quadrilateral.” finds perimeter and explains correctly; incorrect or no conjecture finds correct perimeter, no explanation; correct or close on conjecture finds correct perimeter only Triangle Midsegment Theorem. The left and right sides are both

28. [4]

[3] [2] [1]

BC = 13 by the Perpendicular Bisector Theorem. AE = 5 by the Perpendicular Bisector Theorem. BE = 12 by the Perpendicular Bisector Theorem, so DB = DE + BE = 12 + 12 = 24. ∆DEC ≅ ∆BEC by SAS, so DC = BC = 13. finds three lengths with correct explanations finds two lengths with correct explanations finds one length with correct explanation

OTHER 29. Assume a quadrilateral has more than three obtuse angles. Then it has four angles, each with a measure greater than 90. Their sum is greater than 360, which contradicts the fact that the sum of the measures of the angles of a quadrilateral is 360. Thus a quadrilateral can have no more than three obtuse angles.

2