5.3

Medians and Altitudes in a Triangle

Median• Segment whose endpoints are

a vertex of a triangle and the midpoint of the opposite side.

• All three medians in a triangle intersect at the centroid of the triangle.

Length relationship within a Median

• The centroid is located of the distance from each vertex to opposite midpoint

2

3

S

I

M

E L

PO

2

32

MO ME

MO OE

Ex: Find all segment lengths

S

I

M

E L

PO

:

8

5

6

8.4

Given

MO

OP

EL

OL

:Find

OE

OS

PS

OI

Perimeter OSL

Ex: Find all segment lengths

S

I

M

E L

PO

:

4 10

11

5( 3)

Given

MO x

OE

OS x

:Find

x

MO

ME

OS

OP

Altitude • Perpendicular segment from a

vertex to the line containing the opposite side

• All three altitudes in a triangle intersect at the orthocenter of the triangle.

Altitude (cont.) • Find the orthocenter

Finding the eqn of the line containing a median

• Connects vertex to opposite midpoint• First find midpoint of opp side• Find slope between vertex point and midpoint• Write eqn with point-slope form

Ex: Find eqn of the line containing a median from A to BC

• A (4, 6) B (-2, 8) C (0, 10)

Find the eqn of the line containing an altitude

• Vertex to perp slope of opposite side• First find slope of opp side• Find the perp slope (opp reciprocal)• Write eqn with point-slope form using vertex

and the perp slope

Ex: Find eqn of the line containing an altitude from A to BC

• A (4, 6) B (-2, 8) C (0, 10)

Statements Reasons

Ex: ProofM

BA P

Given : is a median

is an altitude

Prove :

MB

MB

MBA MBP

What is the special name for each segment?

• RZ• SV• SU• XY

mRSV = mTSVRU = UT, SY TY

YZ TS

R

X

V U