Answer:
27 inches, 36 inches, 45 inches
Step-by-step explanation:
The given relation between side lengths can be used with the Pythagorean theorem to find the side lengths.
Three integers that make up the side lengths of a right triangle are referred to as a "Pythagorean triple." One of the first of these that we learn is (3, 4, 5), because ...
3² +4² = 5² ⇔ 9 +16 = 25
This triple has some interesting properties. Among other things, it is the only "primitive" triple that forms an arithmetic sequence. Any other triple that forms an arithmetic sequence must be a multiple of (3, 4, 5).
The (3, 4, 5) sequence has a common difference of 1. The triangle of interest has sides with a common difference of 9. Hence the multiplier must be 9, and the side lengths we seek are 9×(3, 4, 5) = (27, 36, 45).
The side lengths of the triangle are 27 inches, 36 inches, and 45 inches.
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If we let x represent the longer side length, then the other two sides are (x-9) and (x+9). The Pythagorean relation tells us ...
(x -9)² +x² = (x +9)² . . . . . sum of squares of legs is square of hypotenuse
2x² -18x +81 = x² +18x +81 . . . . simplify
x² -36x = 0 . . . . . subtract right-side expression
x(x -36) = 0 ⇒ x = 0 or x = 36 . . . . . . from zero product rule
The value of x must be greater than 9 in this scenario, so the solution is ...
x -9 = 36 -9 = 27
x = 36
x +9 = 36 +9 = 45
The side lengths of the triangle are 27, 36, and 45 inches.
