22. Medians and Altitudes of Triangles

1. 22. Medians and Altitudes of Triangles

2. USING MEDIANS OF A TRIANGLE A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. D

3. USING MEDIANS OF A TRIANGLE centroid centroid centroid acute triangle right triangle obtuse triangle The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle. The centroid is always inside the triangle. The medians of a triangle have a special concurrency property.

4. USING MEDIANS OF A TRIANGLE THEOREM AP= AD 2 2 2 BP = BF 3 3 3 CP= CE THEOREM 5.7 Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two-thirdsof the distance from each vertex to the midpoint of the opposite side. If P is the centroid of ∆ABC, then P

5. USING MEDIANS OF A TRIANGLE The centroid of a triangle can be used as its balancing point. A triangular model of uniformthickness and density willbalance at the centroid of the triangle.

6. Using the Centroid of a Triangle P is the centroid of ∆QRS shown below and PT = 5. Find RT and RP. Because P is the centroid, RP = RT. 2 3 1 Then PT = RT – RP = RT 3 1 Substituting 5 for PT, 5 = RT, so RT = 15. 3 2 2 Then RP = RT = (15) = 10. 3 3 SOLUTION So, RP = 10 and RT= 15.

7. Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find LS. LS = 14

8. Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find NQ. NS + SQ = NQ 12 = NQ

9. In ∆JKL, ZW = 7, and LX = 8.1. Find KW. KW = 21

10. In ∆JKL, ZW = 7, and LX = 8.1. Find LZ. Centroid Thm. Substitute 8.1 for LX. LZ = 5.4 Simplify.

11. Finding the Centroid of a Triangle Find the coordinates of the centroid of ∆JKL. J (7, 10) N (3, 6) P L Choosethe median KN. Findthe coordinates of N, the midpoint of JL. (5, 2) The coordinates of N are K 3 + 7 6 + 10 , = 2 2 10 16 , = (5, 8) 2 2 (5, 8) SOLUTION The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.

12. Finding the Centroid of a Triangle Find the coordinates of the centroid of ∆JKL. J (7, 10) (3, 6) (5, 6) L Determine the coordinates ofthe centroid, which is • 6, or 4 units up from vertex K along the median KN. (5, 2) 2 3 K (5, 8) SOLUTION Findthe distance from vertex Kto midpoint N. The distance fromK (5,2) to N(5,8) is 8 – 2, or 6 units. N P M The coordinates of the centroid P are (5, 2 + 4), or (5, 6) [Yellow coordinates appear.]

13. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle.

14. In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle.

15. Drawing Altitudes and Orthocenters Where is the orthocenter of an acute triangle? SOLUTION Draw an example. The three altitudes intersect at G, a point inside the triangle.

16. Drawing Altitudes and Orthocenters Where is the orthocenter of a right triangle? The two legs, LM and KM, are also altitudes. They intersect at the triangle’s right angle. SOLUTION This implies that the orthocenter is on the triangle at M, the vertex of the right angle of the triangle.

17. THEOREM If AE, BF, and CD are the altitudes of ∆ABC, then the lines AE, BF, and CD intersect at some point H. USING ALTITUDES OF A TRIANGLE Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent.