Medians and Altitudes of Triangles

Concept 37

• Median – a segment that connects the vertex of the triangle to the midpoint of the opposite side of the triangle.

• Median of a Triangle

– Medians do have one vertex as an endpoint.

– Centroid – the point at which medians meet at

one point.

– Where is the centroid located?

• Always inside the triangle.

Medians of a Triangle Theorem

Example 1

The medians of ABC meet at centroid,

point D. Find the indicated values.

Find BG

Find BD

=12

= 8

Example 2

G is the centroid of ABC, AD = 15, CG = 13, and

AD CB .

Find the length of each segment.

a. AG =

b. GD =

c. CD = d. GE =

e. GB = f. Find the perimeter

of ABC

10

5

6.5

13

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

4. In ΔABC, CG = 4. Find GE.

Graph point D.

Find the midpoint D of BC.

Find the centroid of the given triangle.

• Altitude – a segment from a vertex that is perpendicular to the opposite side or to the line containing the opposite side.

Altitudes of a Triangle

– Altitudes have one vertex as an endpoint.

– Orthocenter – the point at which altitudes

meet at one point.

– Where is the orthocenter located?

• Acute Triangle

– Inside triangle

• Right Triangle

– On the vertex of the right angle

• Obtuse Triangle

– Outside, behind the obtuse angle

Altitudes of a Triangle Theorem

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

Altitude through point I.Opposite

reciprocal:

Altitude through point J.

Opposite

reciprocal:

Altitude through point H.

Opposite

reciprocal:

Orthocenter

A. (1, 0)

B. (0, 1)

C. (–1, 1)

D. (0, 0)

7. COORDINATE GEOMETRY The vertices of ΔABCare A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of ΔABC.

Perpendicular

Bisector

Angle

Bisector

Median

Altitude

circumcenter

incenter

centroid

orthocenter