Chapter 5 day 4
- 1. GT Geometry 1/4/13 • Take out the piece of patty paper from yesterday • HW on the corner of your desk • Pass out papers in bin! • Midterm = 2 test grades
- 2. Objectives Apply properties of medians of a triangle. Apply properties of altitudes of a triangle.
- 3. Vocabulary median of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangle
- 4. 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.
- 5. 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.
- 6. 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.
- 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.
- 8. Helpful Hint The height of a triangle is the length of an altitude.
- 9. vocabulary The midsegment of a triangle - Segment that joins the midpoints of any two sides of a triangle.
- 11. Theorem The midsegment of a triangle is half the length of, and parallel to, the third side of a triangle.
- 12. GT Geometry 1. Turn in Cumulative Test on the book shelf, by the time the bell rings, nice neat pile. 2. Clear your desk of everything but your geometric tools and a pencil 3. Pass out papers in bin!
- 13. Examples : Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find LS. Centroid Thm. Substitute 21 for RL. LS = 14 Simplify.
- 14. Examples : Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find NQ. Centroid Thm. NS + SQ = NQ Seg. Add. Post. Substitute NQ for NS. Subtract from both sides. Substitute 4 for SQ. 12 = NQ Multiply both sides by 3.
- 15. Examples : Using the Centroid to Find Segment Lengths In ∆JKL, ZW = 7, and LX = 8.1. Find KW. Centroid Thm. Substitute 7 for ZW. KW = 21 Multiply both sides by 3.
- 16. 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.
- 17. 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 ∆.
- 18. Use the figure for Items 1–3. In ∆ABC, AE = 12, DG = 7, and BG = 9. Find each length. 1. AG 8 2. GC 14 3. GF 13.5 For Items 4 and 5, use ∆MNP with vertices M (– 4, –2), N (6, –2) , and P (–2, 10). Find the coordinates of each point. 4. the centroid (0, 2) 5. the orthocenter