Objectives/Assignments <ul><li>Use proportionality theorems to calculate segment lengths. </li></ul><ul><li>To solve real-life problems, such as determining the dimensions of a piece of land. </li></ul>
Use Proportionality Theorems <ul><li>In this lesson, you will study four proportionality theorems. Similar triangles are used to prove each theorem. </li></ul>
Theorems 6.4 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 =
Theorems 6.5 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. RT TQ RU US = If , then TU ║ QS.
Ex. 1: Finding the length of a segment <ul><li>In the diagram AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC? </li></ul>
<ul><li>Step: </li></ul><ul><li>DC EC </li></ul><ul><li>BD AE </li></ul><ul><li>4 EC </li></ul><ul><li>8 12 </li></ul><ul><li>4(12) </li></ul><ul><li>8 </li></ul><ul><li>6 = EC </li></ul><ul><li>Reason </li></ul><ul><li>Triangle Proportionality Thm. </li></ul><ul><li>Substitute </li></ul><ul><li>Multiply each side by 12. </li></ul><ul><li>Simplify. </li></ul>= = = EC <ul><li>So, the length of EC is 6. </li></ul>
Ex. 2: Determining Parallels <ul><li>Given the diagram, determine whether MN ║ GH. </li></ul>LM MG 56 21 = 8 3 = LN NH 48 16 = 3 1 = 8 3 3 1 ≠ MN is not parallel to GH.
Theorem 6.6 <ul><li>If three parallel lines intersect two transversals, then they divide the transversals proportionally. </li></ul><ul><li>If r ║ s and s║ t and l and m intersect, r, s, and t, then </li></ul>UW WY VX XZ =
Theorem 6.7 <ul><li>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. </li></ul><ul><li>If CD bisects ACB, then </li></ul>AD DB CA CB =
Ex. 3: Using Proportionality Theorems <ul><li>In the diagram 1 2 3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU? </li></ul>
SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8.6 PQ QR ST TU = 9 15 11 TU = 9 ● TU = 15 ● 11 Cross Product property 15(11) 9 55 3 = TU = Parallel lines divide transversals proportionally. Substitute Divide each side by 9 and simplify. <ul><li>So, the length of TU is 55/3 or 18 1/3. </li></ul>
Ex. 4: Using the Proportionality Theorem <ul><li>In the diagram, CAD DAB. Use the given side lengths to find the length of DC. </li></ul>
Solution: <ul><li>Since AD is an angle bisector of CAB, you can apply Theorem 8.7. Let x = DC. Then BD = 14 – x. </li></ul>AB AC BD DC = 9 15 14-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. <ul><li>So, the length of DC is 8.75 units. </li></ul>
Use proportionality Theorems in Real Life <ul><li>Example 5: Finding the length of a segment </li></ul><ul><li>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. </li></ul>
Use proportionality Theorems in Real Life <ul><li>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 </li></ul>DE EF AB BC = <ul><li>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. </li></ul>
Ex. 6: Finding Segment Lengths <ul><li>In the diagram KL ║ MN. Find the values of the variables. </li></ul>
Solution <ul><li>To find the value of x, you can set up a proportion. </li></ul>9 13.5 37.5 - x x = 13.5(37.5 – x) = 9x 506.25 – 13.5x = 9x 506.25 = 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 <ul><li>Since KL ║MN, ∆JKL ~ ∆JMN and </li></ul>JK JM KL MN =
Solution <ul><li>To find the value of y, you can set up a proportion. </li></ul>9 13.5 + 9 7.5 y = 9y = 7.5(22.5) y = 18.75 Write the proportion Cross product property Divide each side by 9.
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