Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Upcoming SlideShare
×

Medians & altitudes of a triangle

3,359 views

Published on

Medians and Altitudes of Triangles

Published in: Education
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

Medians & altitudes of a triangle

1. 1. MEDIANS AND ALTITUDES OF A TRIANGLE B E D G A F E
2. 2. MEDIANS OF A TRIANGLEA 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. 3. CENTROIDS OF THE TRIANGLEThe 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. 4. CENTROIDS P centroid P centroidRIGHT TRIANGLE obtuse triangle You see that the centroid is ALWAYS INSIDE THE TRIANGLE 4
5. 5. T H E ORE M : C ON C URRE NC Y OF M E DI ANS OF A T RI A NG LEThe 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 CIf P is the centroid of P ∆ABC, then FAP = 2/3 AD,BP = 2/3 BF, and ACP = 2/3 CE 5
6. 6. EXERCISE: USING THE CENTROID OF ATRIANGLEP is the centroid of ∆QRS shown below R and PT = 5. Find RT and RP. S P T Q 6
7. 7. EXERCISE: USING THE CENTROID OF A TRIANGLEBecause 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. TThen RP = 2/3 RT = Q2/3 (15) = 10► So, RP = 10, and RT = 15. 7
8. 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. 9. EXERCISE: DRAWING ALTITUDES AND ORTHOCENTERS Where is the orthocenter located in each type of triangle?a. Acute triangleb. Right trianglec. Obtuse triangle 9
10. 10. ACUTE TRIANGLE - ORTHOCENTER B E D G A F E∆ABC is an acute triangle. The three altitudesintersect at G, a point INSIDE the triangle. 10
11. 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. 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. 13. CONCURRENCY OF ALTITUDES OF A TRIANGLE F B H A E D CThe 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