Over Chapter 4

• Name______________

• Special Segments in Triangles

Over Chapter 4

A. 22

B. 10.75

C. 7

D. 4.5

Find y if ΔDEF is an equilateral triangle and mF = 8y + 4.

Over Chapter 4

A. 3.75

B. 6

C. 12

D. 16.5

Find x if mA = 10x + 15, mB = 8x – 18, and mC = 12x + 3.

Use the Perpendicular Bisector Theorems

A. Find BC.

Answer: 8.5

BC = AC Perpendicular Bisector Theorem

BC = 8.5 Substitution

Use the Perpendicular Bisector Theorems

B. Find XY.

Answer: 6

Use the Perpendicular Bisector Theorems

C. Find PQ.

PQ = RQ Perpendicular Bisector Theorem

3x + 1 = 5x – 3 Substitution

1 = 2x – 3 Subtract 3x from each side.

4 = 2x Add 3 to each side.

2 = x Divide each side by 2.

So, PQ = 3(2) + 1 = 7.

Answer: 7

A. 4.6

B. 9.2

C. 18.4

D. 36.8

A. Find NO.

A. 2

B. 4

C. 8

D. 16

B. Find TU.

A. 8

B. 12

C. 16

D. 20

C. Find EH.

Use the Angle Bisector Theorems

A. Find DB.

Answer: DB = 5

DB = DC Angle Bisector Theorem

DB = 5 Substitution

Use the Angle Bisector Theorems

B. Find mWYZ.

Use the Angle Bisector Theorems

Answer: mWYZ = 28

WYZ XYW Definition of angle bisector

mWYZ = mXYW Definition of congruent angles

mWYZ = 28 Substitution

Use the Angle Bisector Theorems

C. Find QS.

Answer: So, QS = 4(3) – 1 or 11.

QS = SR Angle Bisector Theorem

4x – 1 = 3x + 2 Substitution

x – 1 = 2 Subtract 3x from each side.

x = 3 Add 1 to each side.

A. 22

B. 5.5

C. 11

D. 2.25

A. Find the measure of SR.

A. 28

B. 30

C. 15

D. 30

B. Find the measure of HFI.

A. 7

B. 14

C. 19

D. 25

C. Find the measure of UV.

Use the Incenter Theorem

A. Find ST if S is the incenter of ΔMNP.

By the Incenter Theorem, since S is equidistant from the sides of ΔMNP, ST = SU.

Find ST by using the Pythagorean Theorem.

a2 + b2 = c2 Pythagorean Theorem

82 + SU2 = 102 Substitution

64 + SU2 = 100 82 = 64, 102 = 100

Use the Incenter Theorem

Answer: ST = 6

Since length cannot be negative, use only the positive square root, 6. Since ST = SU, ST = 6.

SU2 = 36 Subtract 64 from each side.

SU = ±6 Take the square root of each side.

Use the Incenter Theorem

B. Find mSPU if S is the incenter of ΔMNP.

Since MS bisects RMT, mRMT = 2mRMS. So mRMT = 2(31) or 62. Likewise, mTNU = 2mSNU, so mTNU = 2(28) or 56.

Use the Incenter Theorem

mUPR + mRMT + mTNU = 180 Triangle Angle Sum Theorem

mUPR + 62 + 56 = 180 SubstitutionmUPR + 118 = 180 Simplify.

mUPR = 62 Subtract 118 from each side.

Since PS bisects UPR, 2mSPU = mUPR. This

means that mSPU = mUPR. __12

Answer: mSPU = (62) or 31__12

A. 12

B. 144

C. 8

D. 65

A. Find the measure of GF if D is the incenter of ΔACF.

A. 58°

B. 116°

C. 52°

D. 26°

B. Find the measure of BCD if D is the incenter of ΔACF.

A. 13

B. 11

C. 7

D. –13

In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13.

A. –5

B. 0.5

C. 5

D. 10

In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11.

A. –12.5

B. 2.5

C. 10.25

D. 12.5

In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11).

In the figure, point D is the incenter of ΔABC. What segment is congruent to DG?

___

A. DE

B. DA

C. DC

D. DB

___

___

___

___

A. GCD

B. DCG

C. DFB

D. ADE

In the figure, point D is the incenter of ΔABC. What angle is congruent to DCF?

Use the Centroid Theorem

In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.

Centroid Theorem

YV = 12

Simplify.

Use the Centroid Theorem

Answer: YP = 8; PV = 4

YP + PV = YV Segment Addition

8 + PV = 12 YP = 8

PV = 4 Subtract 8 from each side.

A. LR = 15; RO = 15

B. LR = 20; RO = 10

C. LR = 17; RO = 13

D. LR = 18; RO = 12

In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

Use the Centroid Theorem

In ΔABC, CG = 4. Find GE.

Use the Centroid Theorem

Centroid Theorem

CG = 4

6 = CE

Use the Centroid Theorem

Answer: GE = 2

Segment AdditionCG + GE = CE

Substitution4 + GE = 6

Subtract 4 from each side.GE = 2

A. 4

B. 6

C. 16

D. 8

In ΔJLN, JP = 16. Find PM.