Question

find the missing side lengths. leave your answers as radicals in simplest form.
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)

Explanation:

Problem 15:

Step1: Identify triangle type (45-45-90)

In a 45-45-90 triangle, legs are equal, hypotenuse \( = \text{leg} \times \sqrt{2} \). Here, one leg is \( \frac{3\sqrt{2}}{2} \), so \( y = \frac{3\sqrt{2}}{2} \) (equal leg).

Step2: Find hypotenuse \( x \)

Using hypotenuse formula: \( x = y \times \sqrt{2} = \frac{3\sqrt{2}}{2} \times \sqrt{2} = \frac{3 \times 2}{2} = 3 \).

Step1: Identify triangle type (45-45-90)

Legs are equal, hypotenuse \( = \text{leg} \times \sqrt{2} \). One leg is \( 2\sqrt{2} \), so \( y = 2\sqrt{2} \) (equal leg).

Step2: Find hypotenuse \( x \)

\( x = y \times \sqrt{2} = 2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4 \).

Step1: Identify triangle type (30-60-90)

In 30-60-90, sides are \( s \), \( s\sqrt{3} \), \( 2s \) (hypotenuse). Adjacent to 30° is \( s\sqrt{3} = 3\sqrt{3} \), so \( s = 3 \).

Step2: Find \( y \) (opposite 30°)

\( y = s = 3 \).

Step3: Find hypotenuse \( x \)

\( x = 2s = 6 \).

Answer:

\( y = \frac{3\sqrt{2}}{2} \), \( x = 3 \)

Problem 16: