3.7—Medians and Altitudes of a Triangle

Warm Up

1. What is the name of the point where the angle bisectors of a triangle intersect?

Find the midpoint of the segment with the given endpoints.

2. (–1, 6) and (3, 0)

3. Write an equation of the line containing the points (3, 1) and (2, 10) in point-slope form.

3.7—Medians and Altitudes of a Triangle

Objective: Use properties of ______________ and ___________ of a triangle.

Median of a triangle: a segment whose ____________ are a ________ of a triangle and the midpoint of the ____________ side

Centroid of a triangle:the point of ______________ of the_____________ of a triangle

medians altitudes

endpoints vertexopposite

concurrencymedians

The centroid is __________ inside the triangle. (The centroid is also the balancing point of a triangle.)

always

Examples:

1. D is the centroid of ΔABC, ,ACBE ,CBAB FB = 5, EC = 3, and DF = 2.

a. Find CF.

b. Find CG.

c. Find CD.

d. Find the perimeter of ΔABC.

2. a. Find the midpoint of AC . (Label this point D.)

b. Find the length of the median BD .

c. Find the coordinates of the centroid.

(Its distance from B is ⅔ the length of the median.)

___________ of a triangle: the _____________ segment from a _________ to the __________ side (or to the _________ that contains the opposite side)

________________ of a triangle:the point of ____________ of the altitudes of a triangle

The orthocenter can be:

inside the triangle (acute Δs), on the triangle (right Δs), or outside the triangle (obtuse Δs)

Altitudeperpendicular vertex

opposite line

Orthocenterconcurrency

YZ

YZ

Examples:3. Given the coordinates of the vertices of

ΔXYZ are X(9, –5) Y(–2, 3) Z(4, –6)

a.Find the equation of the line containing the side

b. Find the equation of the line containing the altitude from X to

.

Construct the Centroid of the triangle.

Construct the altitude from each vertex to the opposite side using the perpendicular from a point to a line construction.

Construct the Orthocenter of the triangle.