5 4 Medians and Altitudes Say what Vocabulary

5. 4 Medians and Altitudes Say what? ? ?

Vocabulary… p Concurrent- 3 or more lines, rays, or segments that intersect at the same point p Median of a Triangle – a segment from a vertex to the midpoint of the opposite side p Centroid – point of concurrency of 3 medians p 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

p Orthocenter - the point where the 3 altitudes of a intersect p 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

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

With a partner, do #’s 1 -2 on p. 278

p Practice In Exercises 1– 3, use the diagram. G is the centroid of ABC. 1. If BG = 9, find BF. ANSWER 13. 5 2. If BD = 12, find AD. ANSWER 12 3. If CD = 27, find GC. ANSWER 18

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

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