Lesson 5 2 Apply properties of perpendicular bisectors

Lesson 5 -2 Apply properties of perpendicular bisectors of a ∆ Apply properties of angle bisectors of a ∆

c Vocabulary b Ali a Concurrent Lines Three or more lines intersect at one point. (lines a, b and c) Point of Concurrency The point of intersection of the concurrent lines.

Apply properties of perpendicular bisectors of a ∆ A G B C GA = GB = GC Circumcenter The point of intersection of the perpendicular bisectors. It’s Equidistant from the vertices of the triangle.

EXAMPLE 1 PX, PY and PZ are perpendicular bisectors of ∆ABC Find PC 13. 4 7. 3 P is a the circumcenter, PA = PB = PC = 13. 4

Apply properties of angle bisectors of a ∆ B E F G A D C GD = GE = GF Incenter The point of intersection of the angle bisectors. It’s Equidistant from the sides of the triangle.

EXAMPLE 2 MP and LP are angle bisectors of ∆LMN a) Find the distance from P to MN P is a the incenter, PQ = PR = 5 M R 60 o Q 5 5 P 50 o b) m<PMN = 60 o 2 50 o = 30 o L 20 o N

Worksheet Ans. Key Question 1 Use the diagram for exercises 1 -4 SN, TN and VN are perpendicular bisectors of ∆PQR, Find N is a the circumcenter, NP = NR = NQ Q 3. 95 1) NR = NP = 5. 64 S 2) RV = PV = 5. 47 3) TR = TQ = 3. 95 4) QN = NP = 5. 64 T N 5. 64 P 5. 47 V 4. 03 R

Question 2 Use the diagram for exercises 5 -6 CF and EF are angle bisectors of ∆CDE. 5) Find the distance from F to CD F is a the incenter, FG = FR = 42. 1 R C D 54 o 17 o F 42. 1 6) m<FED G Using ∆CDE m<CED = m<FED = 180 o 92 o 2 – 88 o = 46 o = 92 o E

Question 3 Use the diagram for exercises 7 -8 TJ and SJ are angle bisectors of ∆RST. 7) Find the distance from J to RS J is a the incenter, JK = JX = 8. 37 R 8) m<RTJ Using ∆RST 42 o X m<RTS = 180 o – 70 o = 110 o m<RTJ = 110 o 2 = 55 o J 14 o T K 8. 37 S

Question 4 Tell whether each statement is sometimes true, always true or never true. Support your answer with a sketch. 1) The angle bisectors of a ∆ intersect at a point outside the ∆. Answer: Acute Never Sketch: Right Obtuse

Question 4 Tell whether each statement is sometimes true, always true or never true. Support your answer with a sketch. 2) An angle bisector of a ∆ bisects the opposite side. Answer: sometimes Obtuse Sketch: Does not bisect the opposite side Equilateral or Isosceles Bisects the opposite side

Question 4 Tell whether each statement is sometimes true, always true or never true. Support your answer with a sketch. 3) The perpendicular bisector passes through the opposite vertex. Answer: sometimes Obtuse Does not pass through the opposite vertex Sketch: Equilateral or Isosceles Passes through the opposite vertex

Question 4 Tell whether each statement is sometimes true, always true or never true. Support your answer with a sketch. 4) The incenter of a right ∆ is on the ∆. Answer: never Sketch: Right

Question 4 Tell whether each statement is sometimes true, always true or never true. Support your answer with a sketch. 5) The circumcenter of a scalene ∆ is inside the triangle Answer: Acute scalene sometimes Sketch: Right scalene Obtuse scalene

Question 5 P is the incenter of ∆ABC. Which must be true? A. PA = PB. C. YA = YB B. PX = PY D. AX = BZ B Y Z P A X C