# Lesson 4

Ratios in Right Triangles

• Let’s investigate ratios in the side lengths of right triangles.

### Problem 1

Angle $$B$$ is an acute angle in a right triangle. What is a reasonable approximation for angle $$B$$ if the ratio for the opposite leg divided by the hypotenuse is 0.67?

### Problem 2

Estimate the values to complete the table.

angle adjacent leg $$\div$$ hypotenuse opposite leg $$\div$$ hypotenuse opposite leg $$\div$$ adjacent leg
$$A$$
$$C$$ 0.97 0.26 0.27

### Problem 3

Priya says, “I know everything about a right triangle with a 30 degree angle and a hypotenuse with length 1 cm. Here, look.“

• The other angle is 60 degrees.
• The leg adjacent to the 30 degree angle is 0.866 cm long.
• The side opposite the 30 degree angle is 0.5 cm long.

Han asks, “What would happen if a right triangle with a 30 degree angle has a hypotenuse that is 2 cm instead?“

Help them find the missing angles and side lengths in the new triangle. Explain or show your reasoning.

### Problem 4

Triangle $$ABC$$ is equilateral.

1. What is the value of $$x$$?
2. What is the measure of angle $$B$$
(From Unit 4, Lesson 3.)

### Problem 5

An equilateral triangle has side length 8 units. What is the area?

A:

$$16 \sqrt3$$ square units

B:

24 square units

C:

$$24 \sqrt3$$ square units

D:

32 square units

(From Unit 4, Lesson 3.)

### Problem 6

What is the length of the square’s side?

A:

3 units

B:

$$\frac{6}{\sqrt2}$$ units

C:

$$6 \sqrt2$$ units

D:

12 units

(From Unit 4, Lesson 2.)

### Problem 7

A step has a height of 6 inches. A ramp starts 5 feet away from the base of the step, making a $$5.7^\circ$$ angle with the ground. What can you say about the angle the ramp would make with the ground if the ramp starts closer to the step?

A:

The angle would decrease.

B:

The angle would increase.

C:

The angle would stay the same.

D:

We cannot determine anything about the angle.

(From Unit 4, Lesson 1.)

### Problem 8

The quilt is made of squares with diagonals.

The length of $$BD$$ is 4.

1. Find the length of $$AE$$
2. Find the area of square $$ABCD$$
(From Unit 3, Lesson 12.)