Like this presentation? Why not share!

# 6.6 proportions & similar triangles

## by Jessica Garcia on Jan 31, 2011

• 1,326 views

### Views

Total Views
1,326
Views on SlideShare
1,326
Embed Views
0

Likes
0
24
0

No embeds

### Categories

Uploaded via SlideShare as Microsoft PowerPoint

## 6.6 proportions & similar trianglesPresentation Transcript

• 6.6 Proportions & Similar Triangles
• Objectives/Assignments
• Use proportionality theorems to calculate segment lengths.
• To solve real-life problems, such as determining the dimensions of a piece of land.
• Use Proportionality Theorems
• In this lesson, you will study four proportionality theorems. Similar triangles are used to prove each theorem.
• 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
• 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
• 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.
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
• 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 WY VX XZ =
• Theorem 6.7
• 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
• 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 = 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.
• 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 = 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.
• 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.
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
• Since KL ║MN, ∆JKL ~ ∆JMN and
JK JM KL MN =
• Solution
• To find the value of y, you can set up a proportion.
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.