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5-2 Medians and Altitudes of Triangles

You identified and used perpendicular and angle

bisectors in triangles.

- Identify and use medians in triangles.

- Identify and use altitudes in triangles.

Median

- A median of a triangle is a segment whose

endpoints are a vertex and the midpoint of the

opposite side. - (Goes from vertex to opposite side)

Median

- Every triangle has three medians that are

concurrent. - The point of concurrency of the medians of a

triangle is called the centroid and is always

inside the triangle.

Page 335

Balancing Act

- Balance your triangle on the eraser end of your

pencil. Mark the point. This is the centroid of

the triangle. - Fold your triangle to find the midpoint of each

side of your triangle Connect each vertex to the

midpoint of the opposite side. - Do all three line segments meet at one point?

Centroid

- The point of concurrency of the medians of any

triangle is called the centroid. - The centroid is the center of balance (or center

of gravity) of the triangle.

Centroid

In ?XYZ, P is the centroid and YV 12. Find YP

and PV.

Centroid Theorem

YV 12

Simplify.

YP PV YV Segment Addition 8 PV 12 YP

8 PV 4 Subtract 8 from each side.

Answer YP 8 PV 4

In ?LNP, R is the centroid and LO 30. Find LR

and RO.

A. LR 15 RO 15 B. LR 20 RO 10 C. LR

17 RO 13 D. LR 18 RO 12

In ?JLN, JP 16. Find PM.

A. 4 B. 6 C. 16 D. 8

Altitude (height)

- An altitude of a triangle is a perpendicular

segment drawn from a vertex to the line that

contains the opposite side. - (Vertex to opposite side at a right angle.)

Altitude

- Every triangle has three altitudes. If extended,

the latitudes of a triangle intersect in a common

point called the orthocenter.

Page 337

COORDINATE GEOMETRY The vertices of ?HIJ are

H(1, 2), I(3, 3), and J(5, 1). Find the

coordinates of the orthocenter of ?HIJ.

Point-slope form

Distributive Property

Add 1 to each side.

Point-slope form

Distributive Property

Subtract 3 from each side.

Then, solve a system of equations to find the

point of intersection of the altitudes.

Equation of altitude from J

Multiply each side by 5.

Add 105 to each side.

Add 4x to each side.

Divide each side by 26.

(No Transcript)

(No Transcript)

5-2 Assignment

- Page 340, 5-11, 16-19, 27-30

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