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The sides of a triangle have lengths of x, x + 5, and 25. If the longest side is 25, which of the following values of x would make the triangle right? A. 10 B. 12 C. 15 D. 17

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In a right angle, the sum of the squares of the two shorter sides is equal to the square of the longest side.
Here, the longest side is 25, the shorter sides are x and x+5.

[tex]x^2+(x+5)^2=25^2 \\ x^2+x^2+10x+25=625 \\ 2x^2+10x+25=625 \ \ \ |-625 \\ 2x^2+10x-600=0 \ \ \ |\div 2 \\ x^2+5x-300=0 \\ x^2+20x-15x-300=0 \\ x(x+20)-15(x+20)=0 \\ (x-15)(x+20)=0 \\ x-15=0 \ \lor \ x+20=0 \\ x=15 \ \lor \ x=-20 [/tex]

The length of a side must be a positive number, so x=15.
The answer is C.
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