5.4 Use Medians and Altitudes

1. 5.4 Use Medians and Altitudes Hubarth Geometry

2. Median of a Triangle- is a segment from a vertex to the midpoint of the opposite side. Ex 1. Draw a Median S 4 5 T R 6 Solution S 5 4 R T 3 3

3. Centroid- the three medians of a triangle intersect at one point. This point is called the centroid of the triangle. Intersection of Medians of a Triangle C F D P B A E

4. In RST, Qis the centroid and SQ = 8. Find QWand SW. SQ = SW 2 2 3 3 8= SW 3 2 Multiply each side by the reciprocal, . 12= SW 12 – 8 = 4. SW – SQ = Then QW = Ex 2 Use the Centroid of a Triangle Concurrency of Medians of a Triangle Theorem Substitute 8 for SQ. So, QW = 4 and SW = 12.

5. The distance from vertex G(4, 9)to K(4, 3)is 9–3 = 6 units. So, the centroid is (6) = 4 units down from G on GK. 2 Sketch FGH. Then use the Midpoint Formula to find the midpoint Kof FHand sketch median GK. The correct answer is B. 3 2 + 6 5 + 1 K ( , ) = 2 2 Ex 3 Standardized Test Practice K(4, 3) The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side. The coordinates of the centroid Pare (4, 9 – 4), or (4, 5).

6. Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent. G E D A C F B Orthocenter is the point at which the lines containing the three altitudes of the a triangle intersect. A is the orthocenter of the triangle above.

7. Ex 4 Find the Orthocenter Find the orthocenter Pin an acute, a right, and an obtuse triangle. Right triangle Pis on triangle. Obtuse triangle P is outside triangle. Acute triangle Pis inside triangle.

8. Practice 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. 1. If SC = 2100 feet, findPS andPC. 700 ft, 1400 ft 2. If BT = 1000 feet, find TC andBC. 1000ft, 2000ft 3. If PT = 800 feet, findPA andTA. 1600 ft, 2400ft