5.4 Medians and Altitudes

1. 5.4 Medians and Altitudes

2. Vocabulary… • Concurrent- 3 or more lines, rays, or segments that intersect at the same point • Median of a Triangle – a segment from a vertex to the midpoint of the opposite side • Centroid –point of concurrency of 3 medians • Altitude of a Triangle – the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side – there are 3 in a

3. Orthocenter - the point where the 3 altitudes of a intersect • Theorem 5.8: Concurrency of Medians of a -the medians of a intersect at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side. Theorem 5.9: Concurrency of Altitudes of a -the lines containing the altitudes of a are concurrent

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 = Concurrency of Medians of a Triangle Theorem Substitute 8 for SQ. So, QW = 4 and SW = 12.

5. In Exercises 1–3, use the diagram. Gis the centroid of ABC. ANSWER ANSWER ANSWER 13.5 12 18 • Practice 1. If BG = 9, find BF. 2. If BD = 12, find AD. 3. If CD = 27, find GC.

6. Find the orthocenter Pin an acute, a right, and an obtuse triangle. (Draw 3 altitudes…drop perpendicular lines from vertex to opposite side.) • Find the orthocenter. SOLUTION Right triangle Pis on triangle. Acute triangle Pis inside triangle. Obtuse triangle P is outside triangle.

7. Can you answer these?????? • Look at the and answer the following: • 1. Is BD a median of ABC? • 2. Is BD an altitude ABC? • 3. Is BD a perpendicular bisector? B B • A D C A D C