Chapter 5Section 5.4

Medians and Altitudes

E

H

FG

A B

C

M

A B

C

USING MEDIANS OF A TRIANGLE

A median of a triangle is asegment whose endpoints area vertex of the triangle and themidpoint of the opposite side.

D

USING MEDIANS OF A TRIANGLE

The three medians of a triangle are concurrent. The pointof concurrency is called the centroid of the triangle. The centroid is always inside the triangle.

acute triangle right triangle obtuse triangle

centroid centroid centroid

The medians of a triangle have a special concurrency property.

THEOREM

THEOREM 5.8 Concurrency of Medians of a Triangle

USING MEDIANS OF A TRIANGLE

The medians of a triangle intersect at a point that is two-thirdsof the distance from each vertex to the midpoint of the opposite side.

If P is the centroid of ABC, then

BP = BF 23

PAP = AD 23

CP = CE 23

USING MEDIANS OF A TRIANGLE

The centroid of a trianglecan be used as its balancing point.

A triangular model of uniformthickness and density willbalance at the centroid of the triangle.

Using the Centroid of a Triangle

P is the centroid of QRS shown below and PT = 5. Find RT and RP.

SOLUTION

So, RP = 10 and RT = 15.

Then RP = RT = (15) = 10.23

23

Because P is the centroid,

RP = RT.23

Substituting 5 for PT, 5 = RT, so RT = 15.13

Then PT = RT – RP = RT13

Finding the Centroid of a Triangle

Find the coordinates of the centroid of JKL.

SOLUTION The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.

(5, 8)

Choose the median KN. Findthe coordinates of N, the midpointof JL.

(7, 10)

(3, 6)

(5, 2)

3 + 7

2, 6 + 10

2=

10

2, 16

2= (5, 8)

The coordinates of N are

J

K

L

N

P

Find the coordinates of the centroid of JKL.

SOLUTION

J

K

L

N

(5, 8)

(7, 10)

(3, 6)

(5, 2)

P

M

Find the distance from vertex K

to midpoint N. The distance from

K (5,2) to N(5,8) is 8 – 2, or 6 units.

Determine the coordinates of

the centroid, which is • 6, or

4 units up from vertex K along

the median KN.

23

Finding the Centroid of a Triangle

The coordinates of the centroid P are (5, 2 + 4), or (5, 6)

(5, 6)

An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that containsthe opposite side. An altitude can lie inside, on, or outsidethe triangle.

Every triangle has three altitudes. The lines containing the altitudes are concurrent and intersect at a point called the ortho center of the triangle.

USING ALTITUDES OF A TRIANGLE

Drawing Altitudes and Orthocenters

Where is the ortho center of an acute triangle?

SOLUTION

Draw an example.

The three altitudes intersect at G,a point inside the triangle.

Drawing Altitudes and Orthocenters

The two legs, LM and KM, are also altitudes. They intersect at the triangle’s right angle.

This implies that the orthocenter is on the triangleat M, the vertex of the right angle of the triangle.

Where is the orthocenter of a right triangle?

SOLUTION

Drawing Altitudes and Orthocenters

The three lines that contain the altitudes intersect at W, a point outside the triangle.

Where is the orthocenter of an obtuse triangle?

SOLUTION

THEOREM

USING ALTITUDES OF A TRIANGLE

THEOREM 5.9 Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent.

If AE, BF, and CD are the altitudes

of ABC, then the lines AE, BF, and

CD intersect at some point H.

Median

Angle Bisector

Perpendicular Bisector

Altitude

All Four

PO = 11

2

3ML MP

210

3MP

15 MP

NQ NL LQ 2

3NL NQ

28

3NQ

12 NQ

12 8 LQ

4 LQ