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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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.

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주요 개념

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Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. This relationship, a² + b² = c², is essential for finding unknown side lengths when two sides are known or expressed in terms of each other.

Algebraic Expressions and Variables

Using variables to represent unknown side lengths allows setting up equations based on given relationships. For example, if the middle side is x, the shortest side can be expressed as x - 7, and the hypotenuse as x + 1. This translation from words to algebraic expressions is crucial for solving the problem.

Solving Quadratic Equations

After substituting the expressions into the Pythagorean theorem, the resulting equation is quadratic. Solving quadratic equations involves rearranging terms, factoring or using the quadratic formula, and interpreting the solutions in the context of the problem to find valid side lengths.