Over Chapter 4 Name______________ Special Segments in Triangles

Over Chapter 4 5-Minute Check 5 A.22 B C.7 D.4.5 Find y if ΔDEF is an equilateral triangle and m  F = 8y + 4.

Over Chapter 4 5-Minute Check 2 A.3.75 B.6 C.12 D.16.5 Find x if m  A = 10x + 15, m  B = 8x – 18, and m  C = 12x + 3.

Concept

Example 1 Use the Perpendicular Bisector Theorems A. Find BC. Answer: 8.5 BC= ACPerpendicular Bisector Theorem BC= 8.5Substitution

Example 1 Use the Perpendicular Bisector Theorems B. Find XY. Answer: 6

Example 1 Use the Perpendicular Bisector Theorems C. Find PQ. PQ= RQPerpendicular Bisector Theorem 3x + 1= 5x – 3Substitution 1= 2x – 3Subtract 3x from each side. 4= 2xAdd 3 to each side. 2= xDivide each side by 2. So, PQ = 3(2) + 1 = 7. Answer: 7

Example 1 A.4.6 B.9.2 C.18.4 D.36.8 A. Find NO.

Example 1 A.2 B.4 C.8 D.16 B. Find TU.

Example 1 A.8 B.12 C.16 D.20 C. Find EH.

Concept

Example 3 Use the Angle Bisector Theorems A. Find DB. Answer: DB = 5 DB= DCAngle Bisector Theorem DB= 5Substitution

Example 3 Use the Angle Bisector Theorems B. Find m  WYZ.

Example 3 Use the Angle Bisector Theorems Answer: m  WYZ = 28  WYZ   XYWDefinition of angle bisector m  WYZ= m  XYWDefinition of congruent angles m  WYZ= 28Substitution

Example 3 Use the Angle Bisector Theorems C. Find QS. Answer: So, QS = 4(3) – 1 or 11. QS= SRAngle Bisector Theorem 4x – 1= 3x + 2Substitution x – 1= 2Subtract 3x from each side. x= 3Add 1 to each side.

Example 3 A.22 B.5.5 C.11 D.2.25 A. Find the measure of SR.

Example 3 A.28 B.30 C.15 D.30 B. Find the measure of  HFI.

Example 3 A.7 B.14 C.19 D.25 C. Find the measure of UV.

Concept

Example 4 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. a 2 + b 2 = c 2 Pythagorean Theorem SU 2 = 10 2 Substitution 64 + SU 2 = = 64, 10 2 = 100

Example 4 Use the Incenter Theorem Answer: ST = 6 Since length cannot be negative, use only the positive square root, 6. Since ST = SU, ST = 6. SU 2 = 36Subtract 64 from each side. SU= ±6Take the square root of each side.

Example 4 Use the Incenter Theorem B. Find m  SPU if S is the incenter of ΔMNP. Since MS bisects  RMT, m  RMT = 2m  RMS. So m  RMT = 2(31) or 62. Likewise, m  TNU = 2m  SNU, so m  TNU = 2(28) or 56.

Example 4 Use the Incenter Theorem m  UPR + m  RMT + m  TNU =180Triangle Angle Sum Theorem m  UPR =180Substitution m  UPR =180Simplify. m  UPR =62Subtract 118 from each side. Since PS bisects  UPR, 2m  SPU = m  UPR. This means that m  SPU = m  UPR. __ 1 2 Answer: m  SPU = (62) or 31 __ 1 2

Example 4 A.12 B.144 C.8 D.65 A. Find the measure of GF if D is the incenter of ΔACF.

Example 4 A.58° B.116° C.52° D.26° B. Find the measure of  BCD if D is the incenter of ΔACF.

5-Minute Check 2 A.13 B.11 C.7 D.–13 In the figure, A is the circumcenter of ΔLMN. Find x if m  APM = 7x + 13.

5-Minute Check 1 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.

5-Minute Check 3 A.–12.5 B.2.5 C D.12.5 In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11).

5-Minute Check 4 In the figure, point D is the incenter of ΔABC. What segment is congruent to DG? ___ A.DE B.DA C.DC D.DB ___

5-Minute Check 5 A.  GCD B.  DCG C.  DFB D.  ADE In the figure, point D is the incenter of ΔABC. What angle is congruent to  DCF?

Concept

Example 1 Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify.

Example 1 Use the Centroid Theorem Answer: YP = 8; PV = 4 YP + PV= YVSegment Addition 8 + PV= 12YP = 8 PV= 4Subtract 8 from each side.

Example 1 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.

Example 2 Use the Centroid Theorem In ΔABC, CG = 4. Find GE.

Example 2 Use the Centroid Theorem Centroid Theorem CG = 4 6 = CE

Example 2 Use the Centroid Theorem Answer: GE = 2 Segment AdditionCG + GE = CE Substitution4 + GE = 6 Subtract 4 from each side.GE = 2

Example 2 A.4 B.6 C.16 D.8 In ΔJLN, JP = 16. Find PM.

Concept