Question

select the correct answer.

right triangle def has two 45° angles with the hypotenuse, df, having a length of 8. what is de?

4
4\sqrt{2}
8
8\sqrt{2}

Explanation:

Identify the triangle properties

The given triangle \(DEF\) is a right triangle with two \(45^\circ\) angles, which makes it a \(45^\circ-45^\circ-90^\circ\) special right triangle.

Set up the side length ratio

In a \(45^\circ-45^\circ-90^\circ\) triangle, the ratio of the side lengths is \(1 : 1 : \sqrt{2}\).
Let the leg length \(DE = x\).
The hypotenuse \(DF\) is given by:

$$ DF = x\sqrt{2} $$

Solve for the leg length

Substitute the given hypotenuse length \(DF = 8\) into the equation and solve for \(x\):

$$
LATEXBLOCK0
$$

Thus, the length of \(DE\) is \(4\sqrt{2}\).

Answer:

• 4
• \(4\sqrt{2}\) (Correct answer)
• 8
• \(8\sqrt{2}\)