Medians & altitudes of a triangle
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The three medians of any triangle are concurrent and intersect at the triangle's centroid. The centroid is always inside the triangle and is located 2/3 of the distance from each vertex to the midpoint of the opposite side. The orthocenter, where the three altitudes intersect, is inside an acute triangle, on the right angle vertex of a right triangle, and outside an obtuse triangle.
Medians & altitudes of a triangle
- 1. MEDIANS AND ALTITUDES OF A TRIANGLE B E D G A F E
- 2. MEDIANS OF A TRIANGLE A median of a triangle A is a segments whose endpoints are a vertex of the triangle and the MEDIAN midpoint of the opposite side. In the figure, in ∆ ABC , shown at the right, D is the midpoint of side BC . So, segment AD C is a median of the D B triangle 2
- 3. CENTROIDS OF THE TRIANGLE The three medians of a triangle are concurrent. This means that they meet at a point. The point of concurrency is called the CENTROID OF THE TRIANGLE. CENTROID The centroid, labeled P P in the diagrams in the next few slides are ALWAYS inside the triangle. acute triangle 3
- 4. CENTROIDS P centroid P centroid RIGHT TRIANGLE obtuse triangle You see that the centroid is ALWAYS INSIDE THE TRIANGLE 4
- 5. T H E ORE M : C ON C URRE NC Y OF M E DI ANS OF A T RI A NG LE The medians of a triangle intersect at a B point that is two thirds of the distance from D each vertex to the midpoint of the opposite side. E C If P is the centroid of P ∆ABC, then F AP = 2/3 AD, BP = 2/3 BF, and A CP = 2/3 CE 5
- 6. EXERCISE: USING THE CENTROID OF A TRIANGLE P is the centroid of ∆QRS shown below R and PT = 5. Find RT and RP. S P T Q 6
- 7. EXERCISE: USING THE CENTROID OF A TRIANGLE Because P is the centroid. RP = 2/3 R RT. Then PT= RT – RP = 1/3 RT. Substituting 5 for PT, S 5 = 1/3 RT, so P RT = 15. T Then RP = 2/3 RT = Q 2/3 (15) = 10 ► So, RP = 10, and RT = 15. 7
- 8. EXERCISE: FINDING THE CENTROID OF A TRIANGLE J (7, 10) The coordinates of N are: 10 3+7 , 6+10 = 10 , 16 2 2 2 2 8 N Or (5, 8) 6 L (3, 6) P Find the distance from vertex K to midpoint N. 4 The distance from K(5, M 2) to N (5, 8) is 8-2 or 6 units. 2 K (5, 2) 8
- 9. EXERCISE: DRAWING ALTITUDES AND ORTHOCENTERS Where is the orthocenter located in each type of triangle? a. Acute triangle b. Right triangle c. Obtuse triangle 9
- 10. ACUTE TRIANGLE - ORTHOCENTER B E D G A F E ∆ABC is an acute triangle. The three altitudes intersect at G, a point INSIDE the triangle. 10
- 11. RIGHT TRIANGLE - ORTHOCENTER ∆KLM is a right triangle. The two K legs, LM and KM, are also altitudes. J They intersect at the triangle’s right angle. This implies that the orthocenter M L is ON the triangle at M, the vertex of the right angle of thetriangle. 11
- 12. OBTUSE TRIANGLE - ORTHOCENTER P Z W Y Q X R ∆YPR is an obtuse triangle. The three lines that contain the altitudes intersect at W, a point that is OUTSIDE the triangle. 12
- 13. CONCURRENCY OF ALTITUDES OF A TRIANGLE F B H A E D C 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. 13