7.4 Triangle Proportionality Theorems
- 1. Proportionality Theorems The student is able to (I can): • Use properties of similar triangles to find segment lengths. • Apply proportionality and triangle angle bisector theorems. • Apply triangle angle bisector theorems
- 2. Triangle Proportionality Theorem If a line parallel to a side of a triangle intersects the other two sides then it divides those sides proportionally. S P A C E > > PC SE AP AC PS CE = This ratio is not the same as the ratio between the third sides! AP PC PS SE ≠
- 3. Triangle Proportionality Theorem Converse If a line divides two sides of a triangle proportionally, then it is parallel to the third side. S P A C E > > PC SE AP AC PS CE =
- 4. Two Transversal Proportionality If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. G O D T A C > > > CA DO AT OG =
- 5. Example: Find PE S C O P E 10 14 4 x > >
- 6. Example: Find PE 10x = (4)(14) 10x = 56 S C O P E 10 14 4 10 14 4 x = x 28 3 5 5.6 5 5 x = = = > >
- 8. Example: Verify that (15)(8) = (10)(12)? 120 = 120 Therefore, H O R SE HE OS 15 10 12 8 15 10 ? 12 8 = HE OS
- 10. Example: Solve for x. 6x = (10)(9) 6x = 90 x = 15 > > > x 96 10 10 6 9 x =
- 11. Triangle Angle Bisector Theorem An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle. bisects ∠CAB or CD CA CD DB DB AB CA AB = = DA
- 12. Example: Solve for x.
- 13. Example: Solve for x. 3.5 5 12 5 42 42 8.4 5 AD DC AB BC x x x = = = = =