Question

the hypotenuse of a right triangle is \\(\sqrt{21}\\) in and one leg measures 4 in. find the length of the other leg

select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

\\(\bigcirc\\) a. the a, is exactly \\(\square\\). no approximation is necessary.

\\(\bigcirc\\) b. using radicals, a is exactly \\(\square\\) the a, up to three decimal places, is approximately \\(\square\\).

Explanation:

Step1: Apply Pythagorean theorem

Let \(a\) be the unknown leg, \(b = 4\) and \(c=\sqrt{21}\). By \(a^{2}+b^{2}=c^{2}\), we get \(a^{2}=c^{2}-b^{2}\).

Step2: Substitute values

Substitute \(b = 4\) and \(c=\sqrt{21}\) into \(a^{2}=c^{2}-b^{2}\). So \(a^{2}=(\sqrt{21})^{2}-4^{2}\). Since \((\sqrt{21})^{2}=21\) and \(4^{2} = 16\), then \(a^{2}=21 - 16=5\).

Step3: Solve for \(a\)

Take the square root of both sides. \(a=\sqrt{5}\approx2.236\) (using \(\sqrt{5}\approx2.236\) for the decimal approximation).

Answer:

B. Using radicals, \(a\) is exactly \(\sqrt{5}\). The \(a\), up to three decimal places, is approximately \(2.236\)