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 \\(\square\\) units. the approximate length of the other leg, up to three decimal places, is \\(\square\\) units. (simplify your answers.) b. the exact length of the other leg is \\(\square\\) units. no approximation is necessary. (type an integer or a decimal.) triangle image: right triangle with one leg 17, one angle 45°, and two unknown sides (the other leg and hypotenuse) marked with?

Explanation:

Step1: Identify the triangle type

Since one angle is \(45^{\circ}\) and it is a right - triangle, it is a \(45 - 45-90\) triangle. In a \(45 - 45 - 90\) triangle, the legs are equal. Let the legs be \(a\) and \(a\), and the hypotenuse be \(c\). Given one leg \(a = 17\).

Step2: Find the other leg (exact value)

In a \(45 - 45-90\) triangle, the two legs are equal. So the exact length of the other leg is \(17\) units.

Step3: Find the hypotenuse (exact value)

Using the Pythagorean theorem \(c=\sqrt{a^{2}+a^{2}}\), substituting \(a = 17\), we get \(c=\sqrt{17^{2}+17^{2}}=\sqrt{2\times17^{2}} = 17\sqrt{2}\)

Step4: Find the hypotenuse (approximate value)

Calculate \(17\sqrt{2}\approx17\times1.414 = 24.038\)

Answer:

A. Using radicals, the exact length of the other leg is \(17\) units. The approximate length of the other leg, up to three decimal places, is \(17.000\) units. The exact length of the hypotenuse is \(17\sqrt{2}\) units and its approximate value is \(24.038\) units.
B. The exact length of the other leg is \(17\) units. No approximation is necessary.