7.5 Proportions In Triangles

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Presentation transcript:

7.5 Proportions In Triangles Objective: Use Side-Splitter theorem & Triangle-Angle-Bisector Theorem to calculate segment lengths.

Theorems 7.4 Side-Splitter 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 RU = TQ 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 8 12 4(12) 8 6 = EC Reason Side-Splitter Thm. 8 12 4(12) 8 6 = EC Reason Side-Splitter Thm. Substitute Multiply each side by 12. Simplify. = = EC =

Ex. 2: Determining Parallels Given the diagram, determine whether MN ║ GH. LM 56 8 = = MG 21 3 LN 48 3 = = NH 16 1 8 3 ≠ 3 1 MN is not parallel to GH.

Corollary to Theorem 7-4 If three parallel lines intersect two transversals, then they divide the transversals proportionally. If r ║ s and s║ t and l and m intersect, r, s, and t, then UW VX = WY XZ

Triangle-Angle-Bisector Thm 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 CA = DB CB

Ex. 3: Using Triangle-Angle-Bisector Theorem In the diagram 1  2  3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

9 ● TU = 15 ● 11 Cross Multiply TU PQ ST Parallel lines divide transversals proportionally. = QR TU 9 11 = 15 TU 9 ● TU = 15 ● 11 Cross Multiply 15(11) 55 TU = = 9 3

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

Since AD is an angle bisector of CAB, you can apply Theorem 7. 5 Since AD is an angle bisector of CAB, you can apply Theorem 7.5. Let x = DC. Then BD = 14 – x. Solution: AB BD = AC DC 9 14-X = 15 X

Ex. 4 Continued . . . 9 ● x = 15 (14 – x) 9x = 210 – 15x 24x= 210

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. 9 37.5 - x = 13.5 x 13.5(37.5 – x) = 9x 506.25 – 13.5x = 9x 506.25 = 22.5 x 22.5 = x

Solution To find the value of y, you can set up a proportion. 9 7.5 = 13.5 + 9 y 9y = 7.5(22.5) y = 18.75

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