2.  TEKS Focus:  (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.  (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.  (1)(E) Create and use representations to organize, record, and communicate mathematical ideas.  (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

3. A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Every triangle has three medians, and the medians are concurrent.

4. The point of concurrency of the medians of a triangle is the centroid of the triangle. The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance.

5. 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.

6. The height of a triangle is the length of an altitude. Helpful Hint

7. 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.

9. In ∆LMN, RL = 21. Find LS. Find RS. Example: 1 RL = 21= entire length of median. LS = 2/3 of the median length. LS = (2/3)(21) = 14. RS = 1/3 of the median length. RS = (1/3)(21) = 7.

10. In ∆LMN, SQ = 4. Find NS. Find NQ. Example: 2 SQ is the 1/3 part of the median. NS is the 2/3 part of the median. NS is twice the length of SQ. Therefore, if SQ = 4, then NS = 2(4) = 8. And NQ = 4 + 8 =12. Or NQ = 3(4) = 12.

11. A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance? Example: 3

12. Steps: 1.Find the midpoint of each side of the triangle. 2.Draw the median—the segment from the midpoint to the opposite vertex. 3.The concurrent point of the medians is the centroid (8, 5). Example: 3 continued

13. Find the average of the x-coordinates and the average of the y-coordinates of the vertices of ∆PQR. Make a conjecture about the centroid of a triangle. Example: 4

14. The x-coordinates are 0, 6 and 3. The average is 3. The y-coordinates are 8, 4 and 0. The average is 4. The x-coordinate of the centroid is the average of the x-coordinates of the vertices of the ∆, and the y-coordinate of the centroid is the average of the y-coordinates of the vertices of the ∆. The centroid is at (3, 4). Example: 4 continued

15. Example: 5 Answer A for always, S for sometimes, and N for never: 1. The centroid is _______ located outside of the triangle. 2. The orthocenter is ______ located at the vertex of the right angle if the triangle is a right triangle. 3. The medians ______ form the concurrent point called the centroid. 4. Two of the 3 altitudes are ______ located outside of the triangle if the triangle is an obtuse triangle. 5. The orthocenter is _______ located inside of the triangle. A A S N A

16. Example: 6 Find the orthocenter of a triangle with the given vertices of R(8, 9), S(2, 9), and T(4, 1): Steps: 1.Graph the triangle. 2.Pick a segment and find the slope. Go to the opposite vertex and do the negative reciprocal of the slope you found to make it a  line. 3.Repeat with the other side of the triangle. 4.Repeat with the third side of the triangle. 5.The orthocenter is the point of concurrency of the 3 altitudes at (4, 8). R S T