12.5 Proportions & Similar Triangles

Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally. If TU ║ QS, then RT TQ RU US =

Ex. 1: Finding the length of a segment In the diagram AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC?

Step: DC EC BD AE 4 EC (12) 8 6 = EC Reason Triangle Proportionality Thm. Substitute Multiply each side by 12. Simplify. = = = EC  So, the length of EC is 6.

Ex. 2: Determining Parallels Given the diagram, determine whether MN ║ GH.

Ex. 2: Determining Parallels Given the diagram, determine whether MN ║ GH. LM MG = 8 3 = LN NH = 3 1 = ≠ MN is not parallel to GH.

Theorem 8.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. If CD bisects  ACB, then AD DB CA CB =

Ex. 3: Using Proportionality Theorems In the diagram  1   2   3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8.6 PQ QR ST TU = TU = 9 ● TU = 15 ● 11 Cross Product property 15(11) = TU = Parallel lines divide transversals proportionally. Substitute Divide each side by 9 and simplify.  So, the length of TU is 55/3 or 18 1/3.

Ex. 4: Using the Proportionality Theorem In the diagram,  CAD   DAB. Use the given side lengths to find the length of DC.

Solution: Since AD is an angle bisector of  CAB, you can apply Theorem 8.7. Let x = DC. Then BD = 14 – x. AB AC BD DC = X X = Apply Thm. 8.7 Substitute.

Ex. 4 Continued... 9 ● x = 15 (14 – x) 9x = 210 – 15x 24x= 210 x= 8.75 Cross product property Distributive Property Add 15x to each side Divide each side by 24.  So, the length of DC is 8.75 units.

Use proportionality Theorems in Real Life Example 5: Finding the length of a segment Building Construction: You are insulating your attic, as shown. The vertical 2 x 4 studs are evenly spaced. Explain why the diagonal cuts at the tops of the strips of insulation should have the same length.

Use proportionality Theorems in Real Life Because the studs AD, BE and CF are each vertical, you know they are parallel to each other. Using Theorem 8.6, you can conclude that DE EF AB BC = Because the studs are evenly spaced, you know that DE = EF. So you can conclude that AB = BC, which means that the diagonal cuts at the tops of the strips have the same lengths.

Ex. 6: Finding Segment Lengths In the diagram KL ║ MN. Find the values of the variables.

Solution To find the value of x, you can set up a proportion x x = 13.5(37.5 – x) = 9x – 13.5x = 9x = 22.5 x 22.5 = x Write the proportion Cross product property Distributive property Add 13.5x to each side. Divide each side by 22.5  Since KL ║MN, ∆JKL ~ ∆JMN and JK JM KL MN =

Solution To find the value of y, you can set up a proportion y = 9y = 7.5(22.5) y = Write the proportion Cross product property Divide each side by 9.