Question

question 10
10 pts
g.gsr.6.1 (hc)
find the values of x, y, and z.
image of a triangle with a 45° angle, a 30° angle, a right angle, side 8√2, and segments x, y, z
accessible description
options:
○ x = 8, y = 8√3, z= 16
○ x = 16, y = 16√3, z = 32
○ x = 4, y = 4√3, z = 8
○ x = 8, y = 16, z = 16√3

Explanation:

Step1: Analyze the 45° triangle

In a 45-45-90 triangle, the legs are equal, and the hypotenuse is \( \text{leg} \times \sqrt{2} \). Here, hypotenuse is \( 8\sqrt{2} \), so let the leg (x and the height) be \( a \). Then \( a\sqrt{2}=8\sqrt{2} \), so \( a = 8 \). Thus, \( x = 8 \) (and height \( h = 8 \)).

Step2: Analyze the 30-60-90 triangle

In a 30-60-90 triangle, the sides are in ratio \( 1:\sqrt{3}:2 \) (opposite 30°, 60°, 90°). The height \( h = 8 \) is opposite 30°, so hypotenuse \( z \) (opposite 90°) is \( 2 \times 8 = 16 \). The side \( y \) (opposite 60°) is \( 8\sqrt{3} \).

Step3: Match with options

From Step1, \( x = 8 \); from Step2, \( y = 8\sqrt{3} \), \( z = 16 \). This matches the first option.

Answer:

A. \( x = 8, y = 8\sqrt{3}, z= 16 \) (where A is the first option: \( x = 8, y = 8\sqrt{3}, z= 16 \))