Question

multiple choice 0.63 points find the unknown values of the given triangle. triangle with hypotenuse 15, 45° angle, right angle, legs x (base) and y (height) options: ( x = \frac{15sqrt{3}}{3}, y = \frac{15sqrt{3}}{3} ); ( x = 15sqrt{3}, y = 15sqrt{3} ); ( x = \frac{15sqrt{2}}{2}, y = \frac{15sqrt{2}}{2} ); ( x = 15sqrt{2}, y = 15sqrt{2} )

Explanation:

Step1: Identify triangle type

This is a right - isosceles triangle (one angle is \(45^{\circ}\), right angle, so the other non - right angle is also \(45^{\circ}\)), so the legs \(x\) and \(y\) are equal, and hypotenuse \(h = 15\). For a right - isosceles triangle, if the hypotenuse is \(h\) and the legs are \(a\) (since \(x=y = a\)), the relationship is \(h=a\sqrt{2}\), so \(a=\frac{h}{\sqrt{2}}\).

Step2: Calculate \(x\) and \(y\)

Given \(h = 15\), then \(x=y=\frac{15}{\sqrt{2}}\). Rationalizing the denominator, we multiply the numerator and denominator by \(\sqrt{2}\): \(x = y=\frac{15\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{15\sqrt{2}}{2}\)

Answer:

\(x=\frac{15\sqrt{2}}{2},y = \frac{15\sqrt{2}}{2}\) (the option corresponding to this answer is the one with \(x=\frac{15\sqrt{2}}{2},y=\frac{15\sqrt{2}}{2}\))