# PROPORTIONS AND 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 sides proportionally.

In the diagram above,

if DE ∥ AB, then CE / EB  =  CD / DA.

## Converse of the Triangle Proportionality Theorem

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

In the diagram above,

if CE / EB  =  CD / DA, then DE ∥ AB.

## Theorems on Proportionality

Theorem 1 :

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

In the diagram above,

if r ∥ s and s ∥ t, and l and m intersect r, s and t, then

UW / WY  =  VX / XZ.

Theorem 2 :

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.

## Finding the Length of a Segment

Example 1 :

In the diagram shown below

PQ ∥ ST, QS  =  8, SR  =  4 and PT  =  12

Find the length of TR.

Solution :

By Triangle Proportionality Theorem,

SR / QS  =  TR / PT

Substitute.

4 / 8  =  TR / 12

Simplify.

1 / 2  =  TR / 12

Multiply each side by 12.

12 ⋅ (1 / 2)  =  (TR / 12) ⋅ 12

6  =  TR

So,  the length of TR is 6 units.

Example 2 :

In the diagram shown below KL ∥ MN, find the values of the x and y.

Solution :

Finding the value of x :

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

Write proportion.

9 / 13.5  =  (37.5 - x) / x

By cross product property of proportion,

9x  =  13.5(37.5 - x)

9x  =  506.25 - 13.5x

22.5x  =  506.25

Divide each side by 22.5

x  =  22.5

Finding the value of y :

Since KL ∥ MN and ΔJKL ∼ ΔJMN,

JK / JM  =  KL / MN

JK / (JK + KM)  =  KL / MN

9 (9 + 13.5)  =  7.5 / y

9 / 22.5  =  7.5 / y

By cross product property of proportion,

9y  =  7.5 ⋅ 22.5

9y  =  168.75

Divide each side by 9.

y  =  18.75

## Determining Parallels

Example :

In the diagram shown below, determine whether MN ∥ GH.

Solution :

Begin by finding and simplifying the ratios of the two sides divided by MN.

LM / MG  =  56 / 21  =  8 / 3

LN / NH  =  48 / 16  =  3 / 1

Because 8 / 3  ≠  3 / 1, MN is not parallel to GH.

## Using Proportionality Theorems

Example 1 :

In the diagram shown below,

∠1  ≅  ∠2  ≅  ∠3

PQ  =  9, QR  =  15 and ST  =  11

Find the length of TU.

Solution :

Because corresponding angles are congruent the lines are parallel and we can use Theorem 1 on Proportionality.

Parallel lines divide transversals proportionally.

PQ / QR  =  ST / TU

Substitute.

9 / 15  =  11 / TU

Simplify.

3 / 5  =  11 / TU

By reciprocal property of proportion,

5 / 3  =  TU / 11

Multiply each side by 11.

11 ⋅ (5 / 3)  =  (TU / 11) ⋅ 11

55 / 3  =  TU

So, the length TU is 55 / 3 or 18 units.

Example 2 :

In the diagram shown below, ∠CAD  ≅  ∠DAB. Use the given side lengths to find the length of DC.

Solution :

Since AD is an angle bisector of ∠CAB, we can apply Theorem 2 on Proportionality.

Let x  =  DC.

Then,

BD  =  14 - x

Apply Theorem 2 on Proportionality.

AB / AC  =  BD / DC

Substitute.

9 / 15  =  (14 - x) / x

3 / 5  =  (14 - x) / x

Multiply each side by 5x.

5x ⋅ (3 / 5)  =  [(14 - x) / x] ⋅ 5x

3x  =  70 - 5x

8x  =  70

Divide each side by 8.

x  =  8.75

Hence, the length of DC is 8.75 units.

## Using Proportionality Theorems in Real Life

Example :

We 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 lengths.

Solution :

Because the studs AD, BE and CF are each vertical, we know that they are parallel to each other. Using Theorem 8.6, you can conclude that

DE / EF  =  AB / BC

Because the studs are evenly spaced, we know that

DE  =  EF

So, we can conclude that

AB  =  BC

which means that the diagonal cuts at the tops of the strips have the same lengths.

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