Holt Geometry

5-1 Perpendicular and Angle Bisectors

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Find each measure of MN. Justify

MN = 2.6

Perpendicular Bisector Theorem

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Write an equation to solve for a. Justify

3a + 20 = 2a + 26

Converse of Bisector Theorem

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Find the measures of BD and BC. Justify

BD = 12

BC =24

Converse of Bisector Theorem

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Find the measure of BC. Justify

BC = 7.2

Bisector Theorem

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Write the equation to solve for x. Justify your equation.

3x + 9 = 7x – 17

Bisector Theorem

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Find the measure.

mEFH, given that mEFG = 50°.Justify

m EFH = 25

Converse of the Bisector Theorem

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1).

Perpendicular Bisectors of a triangle…

• bisect each side at a right angle

• meet at a point called the circumcenter

• The circumcenter is equidistant from the 3 vertices of the triangle.

• The circumcenter is the center of the circle that is circumscribed about the triangle.

• The circumcenter could be located inside, outside, or ON the triangle.

C

Angle Bisectors of a triangle…

• bisect each angle• meet at the incenter• The incenter is equidistant from the 3

sides of the triangle.• The incenter is the center of the circle that

is inscribed in the triangle. • The incenter is always inside the circle.

I

Paste-able!

DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.

GC = 13.4

GM = 14.5

MZ and LZ are perpendicular bisectors of ∆GHJ. Find GM

GK = 18.6 JZ = 19.9

Z is the circumcenter of ∆GHJ. GK and JZ

Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).

MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN.

MP and LP are angle bisectors of ∆LMN. Find mPMN.

mPMN = 30

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Medians of triangles:

• Endpoints are a vertex and midpoint of opposite side.• Intersect at a point called the centroid • Its coordinates are the average of the 3 vertices.

• The centroid is ⅔ of the distance from each vertex to the midpoint of the opposite side.

• The centroid is always located inside the triangle.

5-3: Medians and Altitudes

P

A Z

YX

C

B

2 2 2

3 3 3 AP AY BP BZ CP CX

Holt Geometry

5-1 Perpendicular and Angle Bisectors

Altitudes of a triangle:

• A perpendicular segment from a vertex to the line containing the opposite side.

• Intersect at a point called the orthocenter.

• An altitude can be inside, outside, or on the triangle.

5-3: Medians and Altitudes

In ∆LMN, RL = 21 and SQ =4. Find LS.

LS = 14

In ∆LMN, RL = 21 and SQ =4. Find NQ.

12 = NQ

In ∆JKL, ZW = 7, and LX = 8.1. Find KW.

KW = 21

Example 2: Problem-Solving Application

A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance?

Find the average of the x-coordinates and the average of the y-coordinates of the vertices of ∆PQR. Make a conjecture about the centroid of a triangle.

Find the orthocenter of ∆XYZ with vertices X(3, –2), Y(3, 6), and Z(7, 1).

X