GeometryLesson 5 – 2

Medians and Altitudes of Triangles

Objective:Identify and use medians in triangles.Identify and use attitudes in triangles.

Median

Median of a triangleA segment with endpoints at a vertex of a

triangle and the midpoint of the opposite side.

CentroidCentroidThe point of concurrency of the medians of a

triangle.

Centroid TheoremThe medians of a triangle intersect at a point

called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.

Centroid…

x2x

PK + AP = AK

PK + 2(PK) = AK

PK = 5Find AP

BP = 12Find PL.

JC = 15Find JP.

10

6

5

In triangle ABC, Q is the centroid and BE = 9

Find BQ

Find QE

BEBQ3

2

93

2

6

BEQE3

1

93

1

= 3 OR BQ = 2(QE)

6 = 2(QE)3 = QE

Find FQ

Find QC

In triangle ABC, Q is the centroid and FC = 14

)(3

1FCFQ

)15(3

1

= 5

QC = 2(FQ)

QC = 2(5)

QC = 10

In triangle JKL, PT = 2. Find KP.

How do you know that P is the centroid?

KP = 2(PT) = 2(2) = 4

OR

KTKP3

2

PTKPKP 3

2

23

2 KPKP

3

4)(

3

2 KPKP

3

4

3

1KP

KP = 4

In triangle JKL, RP = 3.5 and JP = 9

Find PL

Find PS

PL = 2(RP)= 2(3.5)= 7

JP = 2(PS)9 = 2(PS)

PS = 4.5

A performance artist plans to balance triangular pieces of metal during her next act. When one such triangle is placed on the coordinate plane, its vertices are located at (1, 10) (5, 0) and (9,5). What are the coordinates of the point where the artist should support the triangle so that it will balance.

The balance point of a triangle is the centroid.

Graph the points.

Hint: To make it easier look for a vertical or horizontal line between a midpoint of a side and

vertex.

Find the midpoint of the side(s) that could make a vertical or horizontal line.

Find the midpoint of AB.

Midpoint of AB =

2

010,

2

51= (3, 5)

Let P be the Centroid, where would it be?From the vertex to the centroid is 2/3 of the whole.

)(3

2CDCP

)6(3

2CP

4CP

Count over from C 4 units and that is P

Centroid: (5, 5)

A second triangle has vertices (0,4), (6, 11.5), and (12,1). What are the coordinates of the point where the artist should support the triangle so that it will balance? Explain your reasoning.

Centroid: (6, 5.5)

Altitude

Altitude of a triangleA perpendicular segment from a vertex to

the side opposite that vertex.

Draw a righttriangle and identifyall the altitudes.

OrthocenterOrthocenterThe lines containing the altitudes of a

triangle are concurrent, intersecting at a point called the orthocenter.

Find the orthocenterThe vertices of triangle FGH are F(-2, 4), G(4,4), and H(1, -2). Find the coordinates of the orthocenter of triangle FGH.

Graph the points.

Cont…

Find an equation from F to GH.

Slope of GH. m = 2

New equation is perpendicular to segment GH.Point F (-2, 4) m = -1/2

y = mx + b

b

2

2

14

3 = b

32

1 xy

Cont…

Find an equation from G to FH.Slope of segment FH m = -2

New equation is perpendicular to segment FH.Point G (4, 4) m = 1/2

b 42

14

2 = b

22

1 xy

Cont…

The orthocenter can be found at the intersection of our 2 new equations.

How can we find the orthocenter?

If the orthocenter lies on an exact point of the graphuse the graph to name. If it does not lie on a pointuse systems of equations to find the orthocenter.

32

1 xy

22

1 xy

Cont…

System of equations:

32

1 xy

22

1 xy

22

13

2

1 xx

By substitution.

23 x1 = x

212

1y y = 2.5

Orthocenter(1, 2.5)

SummaryPerpendicular bisector

SummaryAngle bisector

SummaryMedian

SummaryAltitude

Homework

Pg. 337 1 – 10 all, 12 - 20 E, 27 – 30 all, 48 – 54 E