To find out how many miles Elena needs to drive south and then east to get from home to work, we want to use a Pythagorean triple, where the distance she travels south and east forms a right triangle with the hypotenuse being the direct distance from her home to her office (25 miles).
We know that for a right triangle, the Pythagorean theorem states: \[ a^2 + b^2 = c^2 \] where \( c \) is the hypotenuse (25 miles), and \( a \) and \( b \) are the lengths of the other two sides (south and east distances, respectively).
If we look at the possible responses, we should check which one satisfies the equation \( a^2 + b^2 = 25^2 = 625 \).
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7 miles south and 24 miles east: \[ 7^2 + 24^2 = 49 + 576 = 625 \] This is correct.
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5 miles south and 20 miles east: \[ 5^2 + 20^2 = 25 + 400 = 425 \] This does not equal 625.
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7 miles south and 18 miles east: \[ 7^2 + 18^2 = 49 + 324 = 373 \] This does not equal 625.
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5 miles south and 24.5 miles east: \[ 5^2 + 24.5^2 = 25 + 600.25 = 625.25 \] This does not equal 625.
Considering all the options, the only pair that forms a Pythagorean triple and is consistent with the problem is:
7 miles south and 24 miles east.
1 answer