Question

use the pythagorean theorem to calculate the hypotenuse of a right triangle.
$a^2 + b^2 = c^2$
$3^2 + 4^2 = x^2$
$9 + 16 = x^2$
$25 = x^2$
$sqrt{25} = x$
$5\text{ in} = x$
the length of the hypotenuse = \boxed{\quad}\text{in}

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem for a right triangle is \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse. Here, \(a = 3\) in, \(b = 4\) in.

Step2: Substitute values into the formula

Substitute \(a = 3\) and \(b = 4\) into \(a^2 + b^2 = c^2\), we get \(3^2 + 4^2 = x^2\) (let \(x\) be the hypotenuse).

Step3: Calculate the squares

\(3^2 = 9\) and \(4^2 = 16\), so \(9 + 16 = x^2\).

Step4: Sum the values

\(9 + 16 = 25\), so \(25 = x^2\).

Step5: Take the square root

Take the square root of both sides: \(\sqrt{25} = x\), and \(\sqrt{25}=5\).

Answer:

5