Question

r = 3; t = 3 r = 3; t = 3\sqrt{2} r = 3\sqrt{2}; t = 3\sqrt{2} r = 3\sqrt{5}; t = 3\sqrt{5}

Explanation:

Step1: Use the property of a - 45 - 90 triangle

In a 45 - 45 - 90 triangle, the legs are equal (\(r = t\)), and the hypotenuse \(h\) is related to the leg \(x\) by the formula \(h=x\sqrt{2}\). Here, \(h = 3\sqrt{10}\), and let \(x=r=t\).

Step2: Solve for \(x\)

We know that \(h=x\sqrt{2}\), so \(3\sqrt{10}=x\sqrt{2}\). Then \(x=\frac{3\sqrt{10}}{\sqrt{2}}\). Using the rule \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\) (\(a = 10\), \(b = 2\)), we get \(x = 3\sqrt{\frac{10}{2}}=3\sqrt{5}\).

Answer:

\(r = 3\sqrt{5};t = 3\sqrt{5}\)