Question

for the given triangle, find the missing lengths. give an exact answer and, where appropriate, an approximation to three decimal places. find the length of the other leg. select the correct choice below and fill in the answer box(es) to complete your choice. a. using radicals, the exact length of the other leg is \boxed{} units. the approximate length of the other leg, up to three decimal places, is \boxed{} units. (simplify your answers.) b. the exact length of the other leg is \boxed{} units. no approximation is necessary. (type an integer or a decimal )

Explanation:

Step1: Identify the triangle type

This is a \(45 - 45-90\) right - triangle. In a \(45 - 45-90\) triangle, the legs are equal (\(a = b\)) and the hypotenuse \(c=a\sqrt{2}\) (where \(a\) and \(b\) are the lengths of the legs). Given one leg \(a = 9\).

Step2: Find the exact length of the other leg

Since in a \(45 - 45-90\) triangle the legs are equal. Let the legs be \(x\) and \(y\). If \(x = 9\), then \(y=9\) (exact value).

Step3: Find the length of the hypotenuse

Using the formula \(c=a\sqrt{2}\), with \(a = 9\). So \(c = 9\sqrt{2}\).

Step4: Approximate the length of the hypotenuse

Calculate \(9\sqrt{2}\approx9\times1.414 = 12.726\)

Answer:

A. Using radicals, the exact length of the other leg is \(9\) units. The approximate length of the hypotenuse, up to three decimal places, is \(12.726\) units.
B. The exact length of the hypotenuse is \(9\sqrt{2}\) units. No approximation is necessary for the other leg (it is \(9\) units exactly).