Question

Dimensions of a Right Triangle The shortest side of a right triangle is 7 in. shorter than the middle side, while the longest side (the hypotenuse) is 1 in. longer than the middle side. Find the lengths of the sides.
Right triangle with sides labeled x, x minus 7, and x plus 1, showing a right angle at the shortest side.

Right triangle with sides labeled x, x minus 7, and x plus 1, showing a right angle at the shortest side.

Accepted Answer

Assign a variable to represent the length of the middle side. Let the middle side be $$x$$ inches. Express the shortest side in terms of $$x. $$Since it is 7 inches shorter than the middle side, the shortest side is $$x - 7$$ inches. Express the longest side (hypotenuse) in terms of $$x. $$Since it is 1 inch longer than the middle side, the hypotenuse is $$x + 1$$ inches. Use the Pythagorean theorem for right triangles, which states that the sum of the squares of the two shorter sides equals the square of the hypotenuse:  \$$ (x - 7)^2$ + x^2$ = (x + 1)^2$ $$ Expand and simplify the equation, then solve the resulting quadratic equation for $$x. $$After finding $$x$$, substitute back to find the lengths of the shortest and longest sides.

Dimensions of a Right Triangle The shortest side of a right triangle is 7 in. shorter than the middle side, while the longest side (the hypotenuse) is 1 in. longer than the middle side. Find the lengths of the sides.

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Key Concepts

Pythagorean Theorem

Algebraic Expressions and Variables

Solving Quadratic Equations