Question

right triangle relationships and trigonometry
the hypotenuse of a ( 45^{circ}-45^{circ}-90^{circ} ) triangle measures ( 4 mathrm{~cm} ). what is the length of one
leg of the triangle?
( 4 sqrt{2} mathrm{~cm} )
( 4 mathrm{~cm} )
( 2 sqrt{2} mathrm{~cm} )
( 2 mathrm{~cm} )

Explanation:

Step1: Use the property of a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle

In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the ratio of the sides is \(1:1:\sqrt{2}\), where the hypotenuse \(c\) is related to the leg \(a\) (or \(b\)) by the formula \(c = a\sqrt{2}\) (since \(a = b\) in this isosceles right - triangle).
Given \(c = 4\) cm, and \(c=a\sqrt{2}\), we can solve for \(a\).

Step2: Solve for the length of the leg

From \(c=a\sqrt{2}\), we can express \(a=\frac{c}{\sqrt{2}}\).
Substitute \(c = 4\) into the formula: \(a=\frac{4}{\sqrt{2}}\).
Rationalize the denominator: \(a=\frac{4\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\) cm.

Answer:

\(2\sqrt{2}\text{ cm}\)