Question

select the correct answer.

right triangle \\(def\\) has two \\(45^{\circ}\\) angles with the hypotenuse, \\(df\\), having a length of 8. what is \\(de\\)?

\\(\bigcirc\\) 8
\\(\bigcirc\\) \\(8\sqrt{2}\\)
\\(\bigcirc\\) \\(4\sqrt{2}\\)
\\(\bigcirc\\) 4

Explanation:

Identify the properties of the given triangle

The triangle \(DEF\) is a right triangle with two \(45^\circ\) angles, which makes it a \(45^\circ-45^\circ-90^\circ\) isosceles right triangle.
The hypotenuse is \(DF = 8\).

Apply the ratio for a 45-45-90 right triangle

In a \(45^\circ-45^\circ-90^\circ\) triangle, the ratio of the sides is \(1 : 1 : \sqrt{2}\).
Therefore, the relationship between a leg \(s\) and the hypotenuse \(h\) is:

$$ h = s\sqrt{2} $$

Solve for the leg length DE

Substitute \(h = 8\) and solve for \(s\) (where \(s = DE\)):

$$
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$$

Answer:

• (A) 8
• (B) \(8\sqrt{2}\)
• (C) \(4\sqrt{2}\) (Correct answer)
• (D) 4