University:
Felician University
Course:
MATH-231 | Geometry I
Academic year:

2019
Views:

496

Pages:

2
Author:

larbiwxcvbn

Proportions in Triangles The Side-Splitter Theorem The Side-Splitter Theorem states the proportional relationship in a triangle: an altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into two segments. Theorem 63: Side-Splitter Theorem In △ABC, if DE ∥ BC , then DB AD = EC AE . The converse of the theorem is also true: If a line intersects two sides of a triangle and divides those sides proportionally, then the line is parallel to the third side. Example: If 7 2 = x 3 , then 2x=21 and x=10.5. The Triangle Angle-Bisector Theorem The Triangle Angle-Bisector Theorem states that if an angle bisector of a triangle divides the opposite side into two segments, then the ratios of the lengths of the two segments to the lengths of the other two sides of the triangle are equal. In △ABC, if AD bisects ∠BAC, then CD BD = AC AB . **Find x:** 11. x + 12 = 15 x = 3 12. 2x - 2 = 6 2x = 8 x = 4 13. 12 = x + 1 x = 11 **Exercises:** Use the figure at the above to complete each proportion. 1. $\frac{MX}{NC} = \frac{AB}{BC}$ 2. $\frac{SO}{OR} = \frac{SM}{MR}$ 3. $\frac{MO}{OC} = \frac{MN}{CN}$ 4. $\frac{NO}{OC} = \frac{NM}{MC}$ **In ΔABC, AB = 6, BC = 8, and AC = 9.** 14. The bisector of ∠A meets BC at point N. Find BN and CN. $\frac{BN}{CN} = \frac{AB}{AC}$ $\frac{x}{8-x} = \frac{6}{9}$ 9x = 48 - 6x 15x = 48 x = 3.2 BN = 3.2, CN = 8 - 3.2 = 4.8 15. XY || CA. Point X lies on BC such that BX = 2, and Y is on BA. Find BY. $\frac{BY}{AY} = \frac{BX}{XC}$ $\frac{a}{b-a} = \frac{2}{5}$ 5a = 2b - 2a 7a = 2b a = 2b/7 BY = 2b/7