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7.5 Proportions & Similar Triangles. Geometry. Objectives/DFA/HW. Objective: You will use the side splitter & triangle angle bisector theorems to solve problems . DFA : p.475 #12 HW – pp.474-476 (2-36 even). Theorem 7.4 Triangle Proportionality Theorem (Side-splitter Theorem).

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7.5 Proportions & Similar Triangles

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7 5 proportions similar triangles

7.5 Proportions & Similar Triangles

Geometry


Objectives dfa hw

Objectives/DFA/HW

  • Objective:

    • You will use the side splitter & triangle angle bisector theorems to solve problems.

  • DFA: p.475 #12

  • HW – pp.474-476 (2-36 even)


Theorem 7 4 triangle proportionality theorem side splitter theorem

Theorem 7.4 Triangle Proportionality Theorem (Side-splitter Theorem)

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

If TU ║ QS, then

RT

RU

=

TQ

US


Theorem 7 5 converse of the triangle proportionality theorem side splitter theorem

Theorem 7.5 Converse of the Triangle Proportionality Theorem(Side Splitter Theorem)

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

RT

RU

If

, then TU ║ QS.

=

TQ

US


Ex 1 finding the length of a segment

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?


Ex 2 determining parallels

Ex. 2: Determining Parallels

  • Given the diagram, determine whether MN ║ GH.


Theorem 7 6

Theorem 7.6

  • 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

VX

UW

=

XZ

WY


Theorem 7 5

Theorem 7.5

  • 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 proportionality theorems

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

Ex. 3: Using Proportionality Theorems


Ex 4 using the proportionality theorem

In the diagram, CAD  DAB. Use the given side lengths to find the length of DC.

Ex. 4: Using the Proportionality Theorem


Solution

Since AD is an angle bisector of CAB, you can apply Theorem 8.7. Let x = DC. Then BD = 14 – x.

Solution:

AB

BD

=

Apply Thm. 7.5

AC

DC

9

14-X

=

15

X


Ex 4 continued

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


Use proportionality theorems in real life1

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

Use proportionality Theorems in Real Life

DE

AB

=

EF

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.

Ex. 6: Finding Segment Lengths


Solution1

To find the value of x, you can set up a proportion.

Solution

9

37.5 - x

=

Write the proportion

Cross product property

Distributive property

Add 13.5x to each side.

Divide each side by 22.5

13.5

x

13.5(37.5 – x) = 9x

506.25 – 13.5x = 9x

506.25 = 22.5 x

22.5 = x

  • Since KL ║MN, ∆JKL ~ ∆JMN and

JK

KL

=

JM

MN


Solution2

To find the value of y, you can set up a proportion.

Solution

9

7.5

=

Write the proportion

Cross product property

Divide each side by 9.

13.5 + 9

y

9y = 7.5(22.5)

y = 18.75


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