**Worksheet on Similar Triangles :**

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

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

**Problem 1 :**

In the diagram shown below ΔACB ∼ ΔDCE.

a. Write the statement of proportionality.

b. Find ∠CDE.

c. Find DC and AD.

**Problem 2 : **

Color variations in the tourmaline crystal shown below lie along the sides of isosceles triangles. In the triangles each vertex angle measures 52°. Explain why the triangles are similar.

**Problem 3 :**

Use properties of similar triangles to explain why any two points on a line can be used to calculate the slope. Find the slope of the line using both pairs of points shown.

**Problem 4 :**

Low-level aerial photos can be taken using a remote-controlled camera suspended from a blimp. we want to take an aerial photo that covers a ground distance g of 50 meters. Use the proportion f/h = n/g to estimate the altitude h that the blimp should fly at to take the photo. In the proportion, use f = 8 cm and n = 3 cm. These two variables are determined by the type of camera used.

**Problem 5 :**

Find the length of the altitude DG in the diagram shown below.

**Problem 1 :**

In the diagram shown below ΔACB ∼ ΔDCE.

a. Write the statement of proportionality.

b. Find ∠CDE.

c. Find DC and AD.

**Solution (a) : **

DC / AC = CE / CB = ED / BA

**Solution (b) : **

Because ΔACB ∼ ΔDCE,

∠A ≅ ∠CDE

Then, we have

∠CDE = ∠A = 79°

**Solution (c) : **

Write proportion.

ED / BA = DC / AC

Substitute.

3 / 12 = DC / 20

Multiply each side by 20.

20 ⋅ (3 / 12) = (DC / 20) ⋅ 20

Simplify.

5 = DC

Because AD = AC - DC,

AD = 20 - 5

AD = 15

Hence, DC is 5 units and AD is 15 units.

**Problem 2 : **

Color variations in the tourmaline crystal shown below lie along the sides of isosceles triangles. In the triangles each vertex angle measures 52°. Explain why the triangles are similar.

**Solution : **

Because the triangles are isosceles, we can determine that each base angle is 64°. Using the AA Similarity Postulate, we can conclude that the triangles are similar.

**Problem 3 :**

Use properties of similar triangles to explain why any two points on a line can be used to calculate the slope. Find the slope of the line using both pairs of points shown.

**Solution : **

By the AA Similarity Postulate ΔBEC ∼ ΔAFD, so the ratios of corresponding sides are the same.

In particular,

CE / DF = BE / AF

By a property of proportions, we have

CE / BE = DF / AF

The slope of a line is the ratio of the change in y to the corresponding change in x. The ratios CE/BE and DF/AF represent the slopes of BC and AD respectively.

Because the two slopes are equal, any two points on a line can be used to calculate its slope. We can verify this with specific values from the diagram.

Using slope formula,

Slope of BC = (3 - 0) / (4 - 2) = 3 / 2

Slope of AD = [6 - (-3)] / (4 - 2) = 9 / 6 = 3 / 2

**Problem 4 :**

Low-level aerial photos can be taken using a remote-controlled camera suspended from a blimp. we want to take an aerial photo that covers a ground distance g of 50 meters. Use the proportion f/h = n/g to estimate the altitude h that the blimp should fly at to take the photo. In the proportion, use f = 8 cm and n = 3 cm. These two variables are determined by the type of camera used.

**Solution : **

Write proportion.

f / h = n / g

Substitute.

8 / h = 3 / 50

By reciprocal property of proportion,

h / 8 = 50 / 3

Multiply each side by 8.

8 ⋅ (h / 8) = 8 ⋅ (50 / 3)

Simplify.

h ≈ 133

Hence, the blimp should fly at an altitude of about 133 meters to take a photo that covers a ground distance of 50 meters.

**Problem 5 :**

Find the length of the altitude DG in the diagram shown below.

**Solution : **

Find the scale factor of ΔADC to ΔFDE.

AC / FE = (12 + 12) / (8 + 8)

AC / FE = 24 / 16

AC / FE = 3 / 2

Now, because the ratio of the lengths of the altitudes is equal to the scale factor, we can write the following equation.

DB / DG = 3 / 2

Substitute 6 for DB and and solve for DG.

6 / DG = 3 / 2

By reciprocal property of proportion,

DG / 6 = 2 / 3

Multiply each side by 6.

6 ⋅ (DG / 6) = (2 / 3) ⋅ 6

Simplify.

DG = 4

Hence, the length of the altitude DG is 4 units.

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

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