5-1 Bisectors of Triangles5-1 Bisectors of Triangles

You used segment and angle bisectors.

• Identify and use perpendicular bisectors in triangles.

• Identify and use angle bisectors in triangles.

Perpendicular BisectorPerpendicular Bisector Perpendicular bisector is any Perpendicular bisector is any

segment that intersects another segment that intersects another segment at its midpoint segment at its midpoint ANDAND is is perpendicular to that segment.perpendicular to that segment.

J

K

S

R

Perpendicular BisectorPerpendicular Bisector A perpendicular bisector of a side of a A perpendicular bisector of a side of a

triangle is a line perpendicular to a side triangle is a line perpendicular to a side through the midpoint of the side.through the midpoint of the side.

(Perpendicular and bisects one side only)(Perpendicular and bisects one side only)

A

B

C

Perpendicular bisector

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A. Find BC.

Answer: 8.5

BC = AC Perpendicular Bisector Theorem

BC = 8.5 Substitution

B. Find XY.

Answer: 6

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.

DefinitionsDefinitions

Concurrent lines Concurrent lines – three or more – three or more lines intersect at a common point.lines intersect at a common point.

Point of concurrency Point of concurrency – the point – the point where concurrent lines intersect.where concurrent lines intersect.

The point of concurrency is also The point of concurrency is also called the called the circumcentercircumcenter of the of the triangletriangle

Concurrent LinesConcurrent Lines If three or more coplanar lines If three or more coplanar lines

intersect at the same point, they are intersect at the same point, they are concurrent lines.concurrent lines.

The point of intersection is the point The point of intersection is the point of concurrency.of concurrency.

Concurrent linesPoint of concurrency

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Angle BisectorAngle Bisector When an angle bisector is used in a When an angle bisector is used in a

triangle, it is a segment. The angle triangle, it is a segment. The angle bisector cuts the angle in half and bisector cuts the angle in half and goes to the other side.goes to the other side.

A

BC

D

Angle bisector

Page 327

A. Find DB.

Answer: DB = 5

DB = DC Angle Bisector Theorem

DB = 5 Substitution

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.

**Set equal to each other

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A. 12

B. 144

C. 8

D. 65

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

**Use Pythagorean Theorem

A. 58°

B. 116°

C. 52°

D. 26°

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

5-1 Assignment5-1 AssignmentPage 329, 2-30 even, skip 4Page 329, 2-30 even, skip 4