Find the length of the hypotenuse of the right triangle shown here

(1) 5

Explanation

We will use the Pythagorean theorem to find the length of the hypotenuse.  If we know the lengths of any two sides of a right triangle, we can find the length of the third side using the Pythagorean theorem.

We will let \(a\) = 3 and \(b\) = 4, and substitute into the Pythagorean equation to find c

 

This is the Pythagorean equation

\( a^2 + b^2 = c^2 \)

Substitute 3 for a and 4 for b

$3^2 + 4^2 = c^2$

Evaluate each exponential expression

$9 + 16 = c^2$

Do the addition

$25 = c^2$

Reverse the sides of the equation so that \(c^2\) is on the left.

$c^2 = 25$

To find \(c\), we must find a number that, when squared, is 25. There are two such numbers, one positive and one negative; they are the square roots of 25. Since \(c\) represents the length of a side of a triangle, \(c\) cannot be negative. For this reason, we need only find the positive square root of 25 to get \(c\).

The symbol √ is used to indicate the positive square root of a number

$ c= \sqrt{25} $

\(\sqrt{25} = 5\) because 5² = 25

\[ c=5 \]The length of the hypotenuse is 5 in.