6.1 practice a - edl · pn x=+62, pm x=−814 find pk. ... 6.3 practice b ... in exercises 5 and 6,...

Copyright © Big Ideas Learning, LLC Geometry All rights reserved. Resources by Chapter

195

6.1 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–3, tell whether the information in the diagram allows you to conclude that point P lies on the perpendicular bisector of RS, or on the angle bisector of DEF .∠ Explain your reasoning.

1. 2. 3.

In Exercises 4–7, find the indicated measure. Explain your reasoning.

4. AD 5. GJ

6. PQ 7. m DGF∠

8. Write an equation of the perpendicular bisector of the segment with the endpoints ( ) ( )2, 2 and 6, 0 .A B− −

9. Explain how you can use the perpendicular bisector of a segment to draw an isosceles triangle.

10. In a right triangle, is it possible for the bisector of the right angle to be the same line as the perpendicular bisector of the hypotenuse? Explain your reasoning. Draw a picture to support your answer.

R

P

ST

RU

P

S

T

D

E

P

F

208.4

8.4

B

A

C

D

G

H JI

2x + 114x + 5

14Q

P R

S

D

G

E

F

(3x + 8)°(5x − 12)°

Geometry Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 196

6.1 Practice B

30°

60°B

D

CA

3(2x − 8)−x + 25

Name _________________________________________________________ Date _________

In Exercises 1–3, tell whether the information in the diagram allows you to conclude that point P lies on the perpendicular bisector of RS, or on the angle bisector of DEF .∠ Explain your reasoning.

1. 2. 3.

In Exercises 4–6, find the indicated measure. Explain your reasoning.

4. AC 5. m LNM∠ 6. m UTW∠

7. Write an equation of the perpendicular bisector of the segment with the endpoints ( ) ( )3, 7 and 1, 5 .G H − −

8. In the figure, line m is the perpendicular 9. You are installing a fountain in the triangular bisector of .PR Is point Q on line m? Is garden pond shown in the point S on line m? Explain your reasoning. figure. You want to place the fountain the same distance from each side of the pond. Describe a way to determine the location of the fountain using angle bisectors.

D

P

F

E

(0, 5)

xN

M

y

( , )52

52

(5, 0)

K

L

R

PS

Q

Q

D

E

F

P

U

T

W

(2x + 3)°(5x − 24)°

9

9V

10

P

Q

R

S

13

1210

m


6.2 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–3, the perpendicular bisectors of ABC intersect at point G, or the angle bisectors of XYZ intersect at point P. Find the indicated measure. Tell which theorem you used.

1. BG 2. CG 3. PS

In Exercises 4 and 5, find the coordinates of the circumcenter of the triangle with the given vertices.

4. ( ) ( ) ( )6, 0 , 0, 0 , 0, 4J K L 5. ( ) ( ) ( )0, 0 , 4, 0 , 6, 6U V W− −

In Exercises 6 and 7, P is the incenter of QRS. Use the given information to find the indicated measure.

6. 4 8,PJ x= − 7PL x= + 7. 6 2,PN x= + 8 14PM x= − Find .PK Find .PL

8. Draw an obtuse isosceles triangle. Find the circumcenter C. Then construct the circumscribed circle.

9. A cellular phone company is building a tower at an equal distance from three large apartment buildings. Explain how you can use the figure at the right to determine the location of the cell tower.

10. Your friend says that it is impossible for the circumcenter of a triangle to lie outside the triangle. Is your friend correct? Explain your reasoning.

Building 1

Building 2

Building 3

8

9

7

G

A

B

C

9

10G

A

B

5

CS

T U

X

Y

Z

2P

3

Q

SR

L

J

P

K

Q

SR

L

N

MP


201

6.2 Practice B

Name _________________________________________________________ Date __________

In Exercises 1–3, find the indicated measure. Tell which theorem you used.

1. PC 2. AP 3. MP

4. Find the coordinates of the circumcenter of the triangle with the vertices ( ) ( ) ( )4, 12 , 14, 6 , and 6, 2 .A B C −

In Exercises 5 and 6, use the diagram and the given information to find the indicated measures.

5. 6 14,LG x= − 3 22NG x= − + 6. 4 2, 3 2, 2 8GL x GE x GK x= − = + = + Find MG and NG. Find GJ and GE.

7. You are using a rotary sprinkler to water the triangular lawn.

a. Explain how to locate the sprinkler the same distance from each side of the triangular lawn.

b. Explain how to locate the sprinkler the same distance from each vertex of the triangular lawn.

c. Which is closer to vertex B, the incenter or the circumcenter? Explain your reasoning.

8. Explain when the circumcenter of a triangle lies outside the triangle.

9. In the figure at the right, what value of x makes G the incenter of ?JKL

A

B

C

H

D

EP4

6

AB

C

HD

E

P

1415

17

PR

Q

L

N

M

G

G

D

C E

L

J K

A

B

C

J L

K

T

G

SR

12

13

x − 3

R

SQ

LM

P

N

13 11

37

48


6.3 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–3, point Q is the centroid of JKL. Use the given information to find the indicated segment lengths.

1. 21AQ = 2. 72JA = 3. 10KQ =

Find QL and AL. Find JQ and QA. Find QA and KA.

4. Find the coordinates of the centroid of the triangle with the vertices ( ) ( ) ( )6, 8 , 3, 1 , and 0, 3 .A B C− −

In Exercises 5 and 6, tell whether the orthocenter is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.

5. ( ) ( ) ( )1, 5 , 4, 3 , 1, 2Q R S− − − 6. ( ) ( ) ( )4, 6 , 3, 2 , 2, 6L M N− − −

7. Given two vertices and the centroid of a triangle, how many possible locations are there for the third vertex? Explain your reasoning.

8. Given two vertices and the orthocenter of a triangle, how many possible locations are there for the third vertex? Explain your reasoning.

9. The centroid of a triangle is at ( )2, 1− and vertices at ( )3, 5− and ( )7, 4 .− − Find the third vertex of the triangle.

10. The orthocenter of a triangle is at the origin, and two of the vertices of the triangle are at ( )5, 0− and ( )3, 4 . Find the third vertex of the triangle.

11. Your friend claims that it is possible to draw an equilateral triangle for which the circumcenter, incenter, centroid, and orthocenter are not all the same point. Do you agree? Explain your reasoning.

12. Your friend claims that when the median from one vertex of a triangle is the same as the altitude from the same vertex, the median divides the triangle into two congruent triangles. Do you agree? Explain your reasoning.

13. Can the circumcenter and the incenter of an obtuse triangle be the same point? Explain.

J

L

KQ

A

J

LK

Q

A

J

L

K

Q A


6.4 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–5, use the graph of ABC.

1. In ,ABC show that the midsegment ED is

parallel to BC and that 12 .ED BC=

2. Find the coordinates of the endpoints of midsegment ,EF which is opposite .AC

3. Show that EF is parallel to AC and that 12 .EF AC=

4. State the coordinates of the endpoints of midsegment .DF

5. Show that DF is parallel to AB and 12 .DF AB=

In Exercises 6–11, use QRS where A, B, and C are the midpoints of the sides.

6. When 16,AB = what is QS?

7. When 68,SR = what is CA?

8. When 46,SR = what is BR?

9. When 3 1 and 5 4,CA x SR x= − = + what is CA?

10. When 6 and 5 8,QS x CS x= = − what is AB?

11. When 5 2 and 2 5,QR x CB x= + = + what is AR?

12. Your friend claims that because each midsegment is half as long as the corresponding side of the triangle, the perimeter of the midsegment triangle is half the perimeter of the original triangle. Is your friend correct? Explain your reasoning.

13. A building has the shape of a pyramid with a square base. The midsegment parallel to the ground of each triangular face of the pyramid has a length of 58 feet. Find the length of the base the pyramid.

2

y

−2

2 x−1

C

A E B

D

Q A

B

R

C

S

58 ft


211

6.4 Practice B

Name _________________________________________________________ Date __________

In Exercises 1–4, use the graph of ABC.

1. Find the coordinates of the midpoint D of ,AB the midpoint E of ,CB and the midpoint F of .AC

2. Graph the midsegment triangle, .DEF

3. Show that , , and .FD CB FE AB DE AC

4. Show that 1 1 12 2 2, , and .FD CB FE AB DE AC= = =

In Exercises 5–8, use LMN. where U, V, and W are the midpoints of the sides.

5. When 9, what is ?LV UW= 6. When ( )2 5 and 8 ,LU x VW x= − = −

what is ?LM 7. When ( ) ( )22 12 and 4 ,NL x x UW x= + = +

what is ?LV 8. When 2 14 and 13 ,UV y MN y= + = −

what is ?WN

9. The bottom two steps of a stairwell are shown. Explain how to use the given measures to verify that the bottom step is parallel to the floor.

10. Your friend claims that a triangle with side lengths of a, b, and c will have half the area of a triangle with side lengths of 2a, 2b, and 2c. Is your friend correct? Explain your reasoning.

2

y

−3

−5

1 x−2−5

C

A

B

L

VU

M W N

13 in.

13 in.

8 in.

floor

8 in.


215

6.5 Practice A

Name _________________________________________________________ Date __________

In Exercises 1 and 2, list the angles of the given triangle from smallest to largest.

1. 2.

In Exercises 3 and 4, list the sides of the given triangle from shortest to longest.

3. 4.

In Exercises 5 and 6, is it possible to construct a triangle with the given side lengths? Explain.

5. 15, 37, 53 6. 9, 16, 8

7. Write an indirect proof that a triangle has at most one obtuse angle.

8. Describe the possible values of x in 9. List the angles of the given triangle the figure shown. from smallest to largest. Explain your reasoning.

10. The shortest distance between two points is a straight line. Explain this statement in terms of the Triangle Inequality Theorem (Theorem 6.11).

M

L

N

3016

18

ED

F

5x

5x8x

87°

21°

A

B

C

x°

2x°

P

RQ(x − 4)°

A

DCB

(4x − 2)°

(6x + 4)°

(9x + 16)°

F

G

H

2x + 312x + 1

x + 16


6.5 Practice B

Name _________________________________________________________ Date _________

In Exercises 1 and 2, list the angles of the given triangle from smallest to largest.

1. 2.

In Exercises 3 and 4, list the sides of the given triangle from shortest to longest.

3. 4.

5. Write an indirect proof that a right triangle has exactly two acute angles.

6. Is it possible to construct a triangle with side lengths ( )5 2 6 , 3 80,x x− + and2 41 if 9?x x+ = Explain.

7. The figure shows several triangles, with labeled side lengths. Which of the triangles are labeled correctly? Explain.

8. Your friend claims that if you are given the three angle measures of a triangle, you can construct a triangle that obeys the Triangle Inequality Theorem (Theorem 6.11), even if you are not given any of the side lengths. Is your friend correct? Explain your reasoning.

N

M

L

x + 10

x

x + 8

U V

W

2x + 1

x 32

x

Q

R S19° 48°

A

C

B

D52°

115°

A

D

F

C

G

B

E4

9

3

5

512

13

6

2545

10


6.6 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–4, copy and complete the statement with , , or .< > = Explain your reasoning.

1. _____ AC DF 2. _____ m HGI m IGJ∠ ∠

3. 1 _____ 2m m∠ ∠ 4. _____ KL MN

In Exercises 5 and 6, write and solve an inequality for the possible values of x.

5. 6.

In Exercises 7 and 8, write a proof.

7. Given: , TV UW TU VW≅ > 8. Given: 1 2,m m∠ > ∠ B is the midpoint of AC.

Prove: m TVU m WUV∠ > ∠ Prove: AF CF>

9. The figure shows two sliding boards. The slide is the same length in each case, but one is steeper than the other. Can you apply the Hinge Theorem (Theorem 6.12) or the Converse of the Hinge Theorem (Theorem 6.13) in this problem? Explain your reasoning.

TU

WV

A

D E

B

F

C

1 2

1

112

14

110°

115°A B

D E

F

C

H 6 6I J

G

35°19°

K L

NM

2

3

(x + 7)°

(2x − 3)°

30°

20°

6(x + 1)

14x − 10


221

6.6 Practice B

Name _________________________________________________________ Date __________

In Exercises 1–4, copy and complete the statement with , , or .< > = Explain your reasoning.

1. _____ BC DE 2. _____ JI GH

3. 1 _____ 2m m∠ ∠ 4. _____ m U m R∠ ∠

In Exercises 5 and 6, write and solve an inequality for the possible values of x.

5. 6.

7. Use the figure to write a proof.

Given: PQ SR≅

Prove: m PQS m RSQ∠ > ∠

8. Two sailboats started at the same location. Sailboat A traveled 5 miles west, then turned 29° toward the north and continued for 8 miles. Sailboat B first went south for 8 miles, then turned 51° toward the east and continued for 5 miles. Which sailboat was farther from the starting point? Explain your reasoning.

9. How are the Hinge Theorem (Theorem 6.12) and the SAS Congruence Theorem (Theorem 5.5) similar? How are they different? Explain your reasoning.

110°

95°

A

BE

D

F

C 70°

75°55°

80°

G

HJ

I

1 2

5x

3x

K

L

NM

3.15 5

3

4

4

P

Q R U

S T

P

Q

R

S

37°

84°

x + 14

2(3x − 8)

70° 70°

48°66°

3x – 18

2(x + 22)

6.1 practice a - edl · pn x=+62, pm x=−814 find pk. ... 6.3 practice b ... in exercises 5 and 6,...

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