The lengths of two sides of a right triangle are given in the figure. Find the missing side length

(1) 60

Explanation

We will use the Pythagorean theorem to find the missing side length.  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 may substitute 11 for either \( a \) or \( b \), but 61 must be substituted for the length \( c \) of the hypotenuse. If we choose to substitute 11 for \( b \), we can find the unknown side length \( a \) as follows.

This is the Pythagorean equation

\[ a^2+b^2=c^2 \]Substitute 11 for \( b \) and 61 for \( c \)

\[ a^2 + 11^2 = 61^2 \]Evaluate each exponential expression

\[ a^2 + 121 = 3,721 \]To isolate \( a^2 \) on the left side, subtract 121 from both sides.

\[ a^2 + 121 - {\color{green}{121}} = 3,721 - \color{green}{121} \]Do the subtraction

\[ a^2 = 3,600 \]If \( a^2 \) =3,600, then \( a \) must be a square root of 3,600.  Because \( a \) represents a length, it must be the positive square root of 3,600.

\[ a = \sqrt{3,600} \]Use a calculator, if necessary, to find the square root

\[ a=60 \]

The missing side length is 60 ft