1. Median A median of a triangle is a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side. Centroid The point of concurrency of the medians of a triangle is called the centroid. The centroid is also called the center of mass or the balancing point of a triangular region

2. Concept Find the center of mass of your triangle by balancing the triangle on your pencil or finger!

3. Concept

4. Example 1 Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify.

5. Example 1 Use the Centroid Theorem Answer: YP = 8; PV = 4 YP + PV= YVSegment Addition 8 + PV= 12YP = 8 PV= 4Subtract 8 from each side.

6. Example 2 Use the Centroid Theorem In ΔABC, CG = 4. Find GE.

7. Example 2 Use the Centroid Theorem Centroid Theorem CG = 4 6 = CE

8. Example 2 Use the Centroid Theorem Answer: GE = 2 Segment AdditionCG + GE = CE Substitution4 + GE = 6 Subtract 4 from each side.GE = 2

9. Example 1 A.LR = 15; RO = 15 B.LR = 20; RO = 10 C.LR = 17; RO = 13 D.LR = 18; RO = 12 In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

10. Example 2 A.4 B.6 C.16 D.8 In ΔJLN, JP = 16. Find PM.

11. Example 3 Find the Centroid on a Coordinate Plane SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist’s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance? UnderstandYou need to find the centroid of the triangle. This is the point at which the triangle will balance.

12. Example 3 Find the Centroid on a Coordinate Plane SolveGraph the triangle and label the vertices A, B, and C. PlanGraph and label the triangle with vertices at (1, 4), (3, 0), and (3, 8). Use the Midpoint Theorem to find the midpoint of one of the sides of the triangle. The centroid is two-thirds the distance from the opposite vertex to that midpoint.

13. Example 3 Find the Centroid on a Coordinate Plane Graph point D. Find the midpoint D of BC.

14. Example 3 Find the Centroid on a Coordinate Plane Notice that is a horizontal line. The distance from D(3, 4) to A(1, 4) is 3 – 1 or 2 units.

15. The centroid P is the distance. So, the centroid is (2) or units to the right of A. The coordinates are. Example 3 Find the Centroid on a Coordinate Plane P

16. Example 3 Find the Centroid on a Coordinate Plane Answer: The artist should place the pole at the point CheckCheck the distance of the centroid from point D(3, 4). The centroid should be (2) or units to the left of D. So, the coordinates of the centroid is. __ 1 3 2 3

17. Altitude An altitude of a triangle is a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. Acute TriangleObtuse TriangleRight Triangle InteriorExterior On side

18. Concept

19. Example 4 Find the Orthocenter on a Coordinate Plane COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of ΔHIJ.

20. Example 4 Find the Orthocenter on a Coordinate Plane Find an equation of the altitude from The slope of so the slope of an altitude is Point-slope form Distributive Property Add 1 to each side.

21. Example 4 Find the Orthocenter on a Coordinate Plane Point-slope form Distributive Property Subtract 3 from each side. Next, find an equation of the altitude from I to The slope of so the slope of an altitude is –6.

22. Example 4 Find the Orthocenter on a Coordinate Plane Equation of altitude from J Multiply each side by 5. Add 105 to each side. Add 4x to each side. Divide each side by –26. Substitution, Then, solve a system of equations to find the point of intersection of the altitudes.

23. Example 4 Find the Orthocenter on a Coordinate Plane Replace x with in one of the equations to find the y-coordinate. Multiply and simplify. Rename as improper fractions.

24. Example 4 Find the Orthocenter on a Coordinate Plane Answer: The coordinates of the orthocenter of ΔHIJ are

25. Homework p.340 #5 – 10, 16 – 20, 27 – 30