To find the length of the unknown side \( a \) in a right triangle, we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the length of the hypotenuse (the longest side), and \( a \) and \( b \) are the lengths of the other two sides.
Assuming \( 8 , \text{ft} \) and \( \sqrt{73} , \text{ft} \) are the lengths of the two legs, we will determine which is the hypotenuse. The hypotenuse must be the longer side.
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Calculate \( 8^2 \): \[ 8^2 = 64 \]
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Calculate \( (\sqrt{73})^2 \): \[ (\sqrt{73})^2 = 73 \]
Since \( \sqrt{73} , \text{ft} \) is longer than \( 8 , \text{ft} \), it will be treated as the hypotenuse \( c \):
Thus, using the Pythagorean theorem: \[ 8^2 + a^2 = (\sqrt{73})^2 \]
Substituting the values we have: \[ 64 + a^2 = 73 \]
To find \( a^2 \): \[ a^2 = 73 - 64 \] \[ a^2 = 9 \]
Taking the square root of both sides: \[ a = \sqrt{9} = 3 , \text{ft} \]
So, the length of the unknown side \( a \) is:
3 ft.
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