Question

find the missing side lengths. *
1 point
5
45°
b
a
a = 5, b = 5
\\( a = \frac { 5 \sqrt { 3 } } { 3 }, b = \frac { 5 \sqrt { 3 } } { 3 } \\)
\\( a = 5 \sqrt { 2 }, b = 5 \sqrt { 2 } \\)
\\( a = \frac { 5 \sqrt { 2 } } { 2 }, b = \frac { 5 \sqrt { 2 } } { 2 } \\)

Explanation:

Step1: Identify the triangle type

This is 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 \(c\) and the legs \(a\) and \(b\) ( \(a = b\) ) satisfy \(c=a\sqrt{2}\) (or \(a=\frac{c}{\sqrt{2}}\)).

Step2: Calculate the side lengths

Given the hypotenuse \(c = 5\). Using the formula \(a=\frac{c}{\sqrt{2}}\), we rationalize the denominator: \(a=\frac{5}{\sqrt{2}}=\frac{5\sqrt{2}}{2}\). Since \(a = b\) in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, \(b=\frac{5\sqrt{2}}{2}\)

Answer:

\(a=\frac{5\sqrt{2}}{2}, b=\frac{5\sqrt{2}}{2}\) (the fourth option)