Answer:
C. 6
Explanation:
You can try the answers to see which satisfies the Pythagorean theorem:
A. 4² +7² = 16+49 ≠ 117
B. 5² +8² = 25 +64 ≠ 117
C. 6² +9² = 36 +81 = 117 . . . . this (C) is the correct choice
and, for completeness, ...
D. 9² +12² = 81 +144 ≠ 117
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Working out
You can also actually work the problem. The Pythagorean theorem tells you ...
x² + (x+3)² = (√117)²
2x² +6x +9 = 117
2x² +6x = 108 . . . . subtract 9
x² +3x = 54 . . . . . . divide by 2
x(x +3) = 54 . . . . . . factor
Now, you can compare to your memorized times tables, where you find 6×9 = 54, so you know that x=6.
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In case times tables are a challenge, you can continue to complete the square:
x² +3x +1.5² = 54 +1.5²
(x +1.5)² = 56.25 = 7.5² . . . . write as squares
x = 7.5 -1.5 = 6 . . . . . . . . . . . square root, subtract 1.5
The value of x is 6.
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The other solution to the quadratic equation is x=-9, but negative segment lengths make no sense. We acknowledge, then ignore, that extraneous solution.