**Proportions and Similar Triangles Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on proportions and similar triangles.

Before look at the worksheet, if you would like to know the stuff related to proportions and similar triangles,

**Problem 1 :**

In the diagram shown below

PQ ∥ ST, QS = 8, SR = 4 and PT = 12

Find the length of TR.

**Problem 2 :**

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

**Problem 3 : **

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

**Problem 4 :**

In the diagram shown below,

∠1 ≅ ∠2 ≅ ∠3

PQ = 9, QR = 15 and ST = 11

Find the length of TU.

**Problem 5 :**

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

**Problem 6 :**

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.

**Problem 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

Hence, the length of TR is 6 units.

**Problem 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

Add 13.5x to each side.

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

**Problem 3 : **

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.

**Problem 4 :**

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

Hence, the length TU is 55 / 3 or 18⅓ units.

**Problem 5 :**

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

Add 5x to each side.

8x = 70

Divide each side by 8.

x = 8.75

Hence, the length of DC is 8.75 units.

**Problem 6 :**

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.

After having gone through the stuff given above, we hope that the students would have understood how to solve problems on proportions and similar triangles.

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