5-2 Medians and Altitudes of Triangles You identified and used perpendicular and angle bisectors in...

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.

MedianA 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)

F

GH

J

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.

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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 ΔXYZ, P is the centroid and YV = 12. Find YP and PV.

Centroid Theorem

YV = 12

Simplify.

Answer: YP = 8; PV = 4

YP + PV = YV Segment Addition

8 + PV = 12 YP = 8

PV = 4 Subtract 8 from each side.

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.

A. 4

B. 6

C. 16

D. 8

In ΔJLN, JP = 16. Find PM.

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.)

K

L

MN

K

L

MN

Altitude

Every triangle has three altitudes. If extended, the latitudes of a triangle intersect in a common point called the orthocenter.

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COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of ΔHIJ.

Find an equation of the altitude from The slope of

so the slope of an altitude is

Point-slope form

Distributive Property

Add 1 to each side.

Point-slope form

Distributive Property

Subtract 3 from each side.

Next, find an equation of the altitude from I to The

slope of so the slope of an altitude is –6.

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.

Substitution,

Then, solve a system of equations to find the point of intersection of the altitudes.

Replace x with in one of the equations to find the y-coordinate.

Multiply and simplify.

Rename as improper fractions.

Answer: The coordinates of the orthocenter of ΔHIJ are

5-2 Assignment

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