Question

what is the length of each leg of the triangle below?

a. 11
b. 15
c. 1
d. \\(\sqrt{22}\\)
e. \\(11\sqrt{2}\\)
f. \\(\sqrt{11}\\)

Explanation:

Identify the given triangle properties

We are given a right triangle with acute angles of \(45^\circ\) and \(45^\circ\), making it an isosceles right triangle (a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle). The hypotenuse (the side opposite the \(90^\circ\) angle) has a length of \(22\). Let the length of each congruent leg be \(x\).

Apply the Pythagorean Theorem

Using the Pythagorean Theorem knowledge point

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Relate to special right triangle ratios

Alternatively, in a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) special right triangle, the ratio of the sides is \(1 : 1 : \sqrt{2}\).
Therefore, the hypotenuse is \(\sqrt{2}\) times the length of a leg:

$$ \text{leg} \times \sqrt{2} = \text{hypotenuse} $$

$$ x\sqrt{2} = 22 $$

$$ x = \frac{22}{\sqrt{2}} = \frac{22\sqrt{2}}{2} = 11\sqrt{2} $$

Match with the given options

The calculated leg length is \(11\sqrt{2}\), which corresponds to option E.

Answer:

• A. 11
• B. 15
• C. 1
• D. \(\sqrt{22}\)
• E. \(11\sqrt{2}\) (Correct answer)
• F. \(\sqrt{11}\)