Question

13)
15)
17)
16)
18)

Explanation:

Problem 13: Isosceles Right Triangle (45-45-90)

Step 1: Identify Triangle Type

This is a right triangle with one angle \( 45^\circ \), so it's an isosceles right triangle (\( a = c \)), and hypotenuse \( = 8\sqrt{2} \).
For a 45-45-90 triangle, \( \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} \).

Step 2: Calculate Leg Length

Substitute hypotenuse \( = 8\sqrt{2} \):
\( a = c = \frac{8\sqrt{2}}{\sqrt{2}} = 8 \).

Problem 14: 30-60-90 Right Triangle

Step 1: Identify Triangle Type

Right triangle with \( 30^\circ \), so it's a 30-60-90 triangle. The side opposite \( 30^\circ \) is the shorter leg (\( x \)), adjacent to \( 30^\circ \) is the longer leg (\( 12 \)), and hypotenuse (\( c \)).

Step 2: Use 30-60-90 Ratios

In 30-60-90, \( \text{longer leg} = \text{shorter leg} \times \sqrt{3} \), so \( 12 = x \times \sqrt{3} \implies x = \frac{12}{\sqrt{3}} = 4\sqrt{3} \).
Hypotenuse \( c = 2 \times \text{shorter leg} = 2 \times 4\sqrt{3} = 8\sqrt{3} \).

Problem 15: 30-60-90 Right Triangle (Hypotenuse = 3)

Step 1: Identify Triangle Type

Right triangle with \( 60^\circ \), so 30-60-90. Hypotenuse \( = 3 \), shorter leg (\( b \)) opposite \( 30^\circ \), longer leg (\( a \)) opposite \( 60^\circ \).

Step 2: Apply Ratios

Shorter leg: \( b = \frac{\text{hypotenuse}}{2} = \frac{3}{2} = 1.5 \).
Longer leg: \( a = b \times \sqrt{3} = \frac{3}{2}\sqrt{3} \).

Problem 16: 30-60-90 Right Triangle (Shorter Leg = \( 11\sqrt{3} \)?) Wait, Correction: Shorter leg is opposite \( 30^\circ \), so if the side opposite \( 30^\circ \) is \( b \), and the side adjacent (longer leg) is \( 11\sqrt{3} \)? Wait, Re-identify:

Wait, the triangle has \( 30^\circ \), right angle, so:

• Let \( b \) = shorter leg (opposite \( 30^\circ \)), \( 11\sqrt{3} \) = longer leg (opposite \( 60^\circ \)), \( a \) = hypotenuse.

Answer:

Step 1: Triangle Type

Isosceles right triangle, hypotenuse \( = 7 \).

Step 2: Leg Length

\( \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2} \).
Thus, \( m = n = \frac{7\sqrt{2}}{2} \).

Final Answers (Summarized):

1. \( a = 8 \), \( c = 8 \)
2. \( x = 4\sqrt{3} \), \( c = 8\sqrt{3} \)
3. \( b = 1.5 \), \( a = \frac{3\sqrt{3}}{2} \)
4. \( b = 11 \), \( a = 22 \)
5. \( a = 7 \), \( b = 7 \)
6. \( m = n = \frac{7\sqrt{2}}{2} \)