Question

question 7 of 10
which is a true statement about a 45 - 45 - 90 triangle?
a. the hypotenuse is \\( \sqrt { 3 } \\) times as long as either leg.
b. each leg is \\( \sqrt { 3 } \\) times as long as the hypotenuse.
c. the hypotenuse is \\( \sqrt { 2 } \\) times as long as either leg.
d. each leg is \\( \sqrt { 2 } \\) times as long as the hypotenuse.

Explanation:

In a 45 - 45 - 90 triangle, the two legs are of equal length (let the length of each leg be \(a\)). By the Pythagorean theorem (\(c^{2}=a^{2}+b^{2}\), and since \(a = b\) in this triangle, we have \(c^{2}=a^{2}+a^{2}=2a^{2}\), where \(c\) is the hypotenuse). Taking the square root of both sides, we get \(c = \sqrt{2a^{2}}=\sqrt{2}a\). This means the hypotenuse \(c\) is \(\sqrt{2}\) times as long as either leg \(a\).

• Option A: The hypotenuse is not \(\sqrt{3}\) times the leg (we derived it is \(\sqrt{2}\) times), so A is wrong.
• Option B: Each leg is \(\frac{1}{\sqrt{2}}\) (or \(\frac{\sqrt{2}}{2}\)) times the hypotenuse, not \(\sqrt{3}\) times, so B is wrong.
• Option C: Matches our derivation that \(c=\sqrt{2}a\), so C is correct.
• Option D: Each leg is \(\frac{\sqrt{2}}{2}\) times the hypotenuse, not \(\sqrt{2}\) times, so D is wrong.

Answer:

C. The hypotenuse is \(\sqrt{2}\) times as long as either leg.