**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

So, 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

So, 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

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

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