Refer to the right triangle shown here. Find the missing side length. Give the exact answer and an approximation to the nearest hundredth.

(1) 5.66

Explanation

Strategy 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 2 for \( a \) either or \( b \), but 6 must be substituted for the length of the hypotenuse. If we choose to substitute 2 for , we can find the unknown side length as follows.

This is the Pythagorean equation

\[ a^2 + b^2 = c^2 \]Substitute 2 for \( a \) and 6 for \( b \)

\[ 2^2 + b^2 = 6^2 \]Evaluate each exponential expression

\[ 4 + b^2 = 36 \]To isolate \( b^2 \) on the left side, undo the addition of 4 by subtracting 4 from both  sides.

\[ 4 + b^2 - {\color{red}{4}} = 36 - {\color{red}{4}} \]Do the subtraction

\[ b^2 = 32 \]We must find a number that, when squared, is 32. Since \( b \) represents the length of aside of a triangle, we consider only the positive square root.

This is the exact legnth

\[ b = \sqrt{32} \]The missing side length is exactly \( \sqrt{32} \) inches long. Since 32 is not a perfect square, its square root is not a whole number.  We can use a calculator to approximate \( \sqrt{32} \).  To the nearest hundredth, the missing side length is 5.66 inches.

\[ \sqrt{32} \approx 5.66 \text{ in.} \]