Question

the diagrams below show measurements of special right triangles. select the correct measure from each pull-down menu to label selected measures in the diagram.

(diagrams of special right triangles: one 45-45-90 triangle with hypotenuse (8sqrt{2}) and a pull-down menu, one 30-60-90 triangle with hypotenuse 10 and two pull-down menus)

Explanation:

Step1: Analyze the 45-45-90 triangle

In a 45-45-90 triangle, the legs are equal, and the hypotenuse \( c = l\sqrt{2} \), where \( l \) is the leg length. Given hypotenuse \( 8\sqrt{2} \), solve for \( l \):
\( l\sqrt{2} = 8\sqrt{2} \) ⇒ \( l = 8 \). So the leg (the missing measure) is 8.

Step2: Analyze the 30-60-90 triangle

In a 30-60-90 triangle, sides are in ratio \( 1 : \sqrt{3} : 2 \) (opposite 30°, 60°, hypotenuse). Hypotenuse = 10, so:

• Side opposite 30° (shorter leg) = \( \frac{10}{2} = 5 \).
• Side opposite 60° (longer leg) = \( 5\sqrt{3} \).

Answer:

• For the 45-45-90 triangle: The missing leg is \( \boldsymbol{8} \).
• For the 30-60-90 triangle: Shorter leg (opposite 30°) is \( \boldsymbol{5} \), longer leg (opposite 60°) is \( \boldsymbol{5\sqrt{3}} \), hypotenuse is 10 (given).

(Note: The specific dropdowns would be filled with these values based on their positions. For the first triangle’s dropdown, select 8; for the 30-60-90 triangle’s shorter leg dropdown, select 5, and longer leg dropdown, select \( 5\sqrt{3} \).)