To solve the problem using the Pythagorean theorem, we can denote the southward distance as \( a \) and the eastward distance as \( b \), with the distance from Elena's home to her office being the hypotenuse \( c \) of a right triangle.
According to the problem, the hypotenuse \( c \) is 25 miles. We need to find integer values for \( a \) and \( b \) that satisfy the equation:
\[ a^2 + b^2 = c^2 \] \[ a^2 + b^2 = 25^2 \] \[ a^2 + b^2 = 625 \]
Now, we will check the given options to see if they satisfy this equation:
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7 miles south and 18 miles east: \[ 7^2 + 18^2 = 49 + 324 = 373 \quad (\text{not 625}) \]
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5 miles south and 20 miles east: \[ 5^2 + 20^2 = 25 + 400 = 425 \quad (\text{not 625}) \]
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7 miles south and 24 miles east: \[ 7^2 + 24^2 = 49 + 576 = 625 \quad (\text{correct}) \]
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5 miles south and 24.5 miles east: \[ 5^2 + 24.5^2 = 25 + 600.25 = 625.25 \quad (\text{not 625}) \]
The only option that correctly uses a Pythagorean triple to find the distances from Elena's home to her office is:
7 miles south and 24 miles east.
1 answer