Question

select the correct answer.
. right triangle def has two 45° angles with the hypotenuse, df, having a length of 8. what is de?
○ a. 8√2
○ b. 4
○ c. 4√2
○ d. 8

Explanation:

Step1: Identify the triangle type

A right - triangle with two \(45^{\circ}\) angles is a \(45 - 45-90\) triangle. In a \(45 - 45-90\) triangle, the ratio of the sides is \(1:1:\sqrt{2}\), where the hypotenuse \(c\) is related to the legs \(a\) (or \(b\)) by the formula \(c = a\sqrt{2}\) (if \(a = b\)).

Step2: Solve for the leg length

Given \(c=DF = 8\), and using the formula \(c=a\sqrt{2}\) (let \(a = DE\)). We can solve for \(a\) by rearranging the formula: \(a=\frac{c}{\sqrt{2}}\). Substitute \(c = 8\) into the formula: \(DE=\frac{8}{\sqrt{2}}\). Rationalize the denominator: \(\frac{8}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{8\sqrt{2}}{2}=4\sqrt{2}\).

Answer:

C. \(4\sqrt{2}\)