Question

for the given triangle, find the missing lengths. give an exact answer and, where appropriate, an approximation to three decimal places.

a. using radicals, the exact length of the other leg is units. the approximate length of the other leg, up to three decimal places, is units
(simplify your answers.)

b. the exact length of the other leg is 9 units. no approximation is necessary.
(type an integer or a decimal.)

find the length of the hypotenuse. select the correct choice below and fill in the answer box(es) to complete your choice.

a. using radicals, the exact length of the hypotenuse is units. the approximate length of the hypotenuse, up to three decimal places, is units
(simplify your answers.)

b. the exact length of the hypotenuse is units. no approximation is necessary.
(type an integer or a decimal.)

Explanation:

Step1: Find the other leg

In a right - angled triangle with one angle equal to \(45^{\circ}\), the two legs are equal. Let the legs be \(a\) and \(b\), and the hypotenuse be \(c\). Using the property of a \(45 - 45-90\) triangle (or Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), when \(a = b\)). Since one leg \(a = 9\), the other leg \(b=9\) (because in a \(45 - 45-90\) triangle, the legs are congruent).

Step2: Find the hypotenuse

Using the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), with \(a = 9\) and \(b = 9\). Then \(c^{2}=9^{2}+9^{2}=81 + 81=162\). So \(c=\sqrt{162}=\sqrt{81\times2}=9\sqrt{2}\approx9\times1.414 = 12.726\)

Answer:

The exact length of the other leg is \(9\) units. The exact length of the hypotenuse is \(9\sqrt{2}\) units and the approximate length of the hypotenuse is \(12.726\) units.