Question

what is the value of \\( c \\) in the figure given below?
right triangle with legs 3 ft and 7 ft, hypotenuse c
\\( \boldsymbol{\text{a}} \\) \\( \sqrt{56} \\) ft
\\( \boldsymbol{\text{b}} \\) \\( \sqrt{58} \\) ft
\\( \boldsymbol{\text{c}} \\) \\( \sqrt{60} \\) ft
\\( \boldsymbol{\text{d}} \\) \\( \sqrt{62} \\) ft

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle (has a right angle). So we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\).
Here, \(a = 3\) ft and \(b=7\) ft, and \(c\) is the hypotenuse (the side we need to find, labeled as \(c\) in the figure).

Step2: Apply the Pythagorean theorem

Substitute \(a = 3\) and \(b = 7\) into the formula \(c^{2}=a^{2}+b^{2}\).
We get \(c^{2}=3^{2}+7^{2}\).
Calculate \(3^{2}=9\) and \(7^{2}=49\). Then \(c^{2}=9 + 49\).
\(9+49 = 58\), so \(c^{2}=58\).
To find \(c\), take the square root of both sides: \(c=\sqrt{58}\) ft.

Answer:

B. \(\sqrt{58}\) ft