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Chapter 5.4 Notes Use Medians and Altitudes

- Goal You will use medians and altitudes of

triangles.

- Medians
- A median of a triangle is a segment from a vertex

to the midpoint of the opposite side. - The three medians of a triangle are concurrent.
- The point of concurrency, called the centroid, is

inside the triangle.

- Theorem 5.8 Concurrency of Medians of a Triangle
- The medians of a triangle intersect at a point

that is two thirds of the distance from each

vertex to the midpoint of the opposite side. - Ex.1 In ?RST, Q is the centroid and SQ 8. Find

QW and SW.

- Ex.2 In ?HJK, P is the centroid and JP 12.

Find PT and JT. - Ex.3 The vertices of ?FGH are F(2, 5), G(4, 9),

and H(6, 1). What is the coordinates of the

centroid P of ?FGH?

- Ex.4 There are three paths through a triangular

park. Each path goes from the midpoint of one

edge to the opposite corner. The paths meet at

point P. - a. If SC 2100 ft, find PS and PC.
- b. If BT 1000 ft, find TC and BC.
- c. If PT 800 ft, find PA and TA.

- Altitudes
- An altitude of a triangle is the perpendicular

segment from a vertex to the opposite side or to

the line that contains the opposite side. - Theorem 5.9 Concurrency of Altitudes of a

Triangle - The lines containing the altitudes of a triangle

are concurrent.

- Concurrency of Altitudes
- The point at which the lines containing the three

altitudes of a triangle intersect is called the

orthocenter of the triangle. - Ex.5 Find the orthocenter P in an acute, a

right, and an obtuse triangle. - Isosceles Triangles
- In an isosceles triangle, the perpendicular

bisector, angle bisector, median, and altitude

from the vertex angle to the base are all the

same segment.

- In an equilateral triangle, this is true for the

special segment from any vertex. - Ex.6 Prove that the median to the base of an

isosceles triangle is an altitude. - Ex.7 Copy the triangle in example 6 and find its

orthocenter. - Ex.8 The vertices of ?ABC are A(1, 5), B(5, 7),

and C(9, 3). What are the coordinates of the

centroid of ?ABC?

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