Medians and Altitudes of a Triangle

1. Medians and Altitudes of a Triangle Geometry Mrs. Spitz Fall 2004

2. Objectives: • Use properties of medians of a triangle • Use properties of altitudes of a triangle P. Spitz – Taos H.S.

3. A median of a triangle is a segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. For instance in ∆ABC, shown at the right, D is the midpoint of side BC. So, AD is a median of the triangle Medians of a triangle P. Spitz – Taos H.S.

4. The three medians of a triangle are concurrent (they meet). The point of concurrency is called the CENTROID OF THE TRIANGLE. The centroid, labeled P in the diagrams in the next few slides are ALWAYS inside the triangle. Centroids of the Triangle P. Spitz – Taos H.S.

5. CENTROIDS - ALWAYS INSIDE THE TRIANGLE P. Spitz – Taos H.S.

6. 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. If P is the centroid of ∆ABC, then AP = 2/3 AD, BP = 2/3 BF, and CP = 2/3 CE Theorem:Concurrency of Medians of a Triangle P. Spitz – Taos H.S.

7. So what? The centroid of a triangle can be used as its balancing point. Try it. Draw and cut out a triangle. Construct the medians of the triangles in order to great the centroid in the middle. Then use your pencil to balance your triangle. If it doesn’t balance, you didn’t construct it correctly. P. Spitz – Taos H.S.

8. P is the centroid of ∆QRS shown below and PT = 5. Find RT and RP. Ex. 1: Using the Centroid of a Triangle P. Spitz – Taos H.S.

9. Because P is the centroid. RP = 2/3 RT. Then PT= RT – RP = 1/3 RT. Substituting 5 for PT, 5 = 1/3 RT, so RT = 15. Then RP = 2/3 RT = 2/3 (15) = 10 ► So, RP = 10, and RT = 15. Ex. 1: Using the Centroid of a Triangle P. Spitz – Taos H.S.

10. Find the coordinates of the centroid of ∆JKL You know that the centroid is two thirds of the distance from each vertex to the midpoint of the opposite side. Choose the median KN. Find the coordinates of N, the midpoint of JL. Ex. 2: Finding the Centroid of a Triangle P. Spitz – Taos H.S.

11. The coordinates of N are: 3+7 , 6+10 = 10 , 16 2 2 2 2 Or (5, 8) Find the distance from vertex K to midpoint N. The distance from K(5, 2) to N (5, 8) is 8-2 or 6 units. Ex. 2: Finding the Centroid of a Triangle P. Spitz – Taos H.S.

12. Determine the coordinates of the centroid, which is 2/3 ∙ 6 or 4 units up from vertex K along median KN. ►The coordinates of centroid P are (5, 2+4), or (5, 6). Ex. 2: Finding the Centroid of a Triangle P. Spitz – Taos H.S.

13. Objective 2: Using altitudes of a triangle An altitude of a triangle is the perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle. Every triangle has 3 altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle. P. Spitz – Taos H.S.

14. Ex. 3: Drawing Altitudes and Orthocenters • Where is the orthocenter located in each type of triangle? • Acute triangle • Right triangle • Obtuse triangle P. Spitz – Taos H.S.

15. Acute Triangle - Orthocenter ∆ABC is an acute triangle. The three altitudes intersect at G, a point INSIDE the triangle. P. Spitz – Taos H.S.

16. Right Triangle - Orthocenter ∆KLM is a right triangle. The two legs, LM and KM, are also altitudes. They intersect at the triangle’s right angle. This implies that the ortho center is ON the triangle at M, the vertex of the right angle of the triangle. P. Spitz – Taos H.S.

17. Obtuse Triangle - Orthocenter ∆YPR is an obtuse triangle. The three lines that contain the altitudes intersect at W, a point that is OUTSIDE the triangle. P. Spitz – Taos H.S.

18. The lines containing the altitudes of a triangle are concurrent. If AE, BF, and CD are altitudes of ∆ABC, then the lines AE, BF, and CD intersect at some point H. Theorem: Concurrency of Altitudes of a Triangle P. Spitz – Taos H.S.