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Arturo is building a flower bed in the shape of a right triangle. The hypotenuse of the right triangle is 13 feet. One of the legs needs to be 7 feet longer than the other leg.



Which equation can be used to find $x$, the length of the shorter leg?


A$x(x+7)=13$
B$x(x+7)=169$
C$x^2+(x+7)^2=13$
D$x^2+(x+7)^2=169$

Verified Answer :

  • Identify the legs as \(x\) and \(x + 7\) and the hypotenuse as \(13\).
  • Apply the Pythagorean theorem \(a^2 + b^2 = c^2\) to the given dimensions.
  • Substitute the side lengths into the theorem to get \(x^2 + (x + 7)^2 = 13^2\).
  • Calculate the square of the hypotenuse to find the final equation: \(\boxed{x^2 + (x + 7)^2 = 169}\).

Explanation

  1. Analysis and Problem Setup

Arturo's flower bed is a right triangle with a hypotenuse of \(13\) feet. Let the length of the shorter leg be \(x\). According to the problem, the other leg is \(7\) feet longer than the shorter leg, meaning its length is \(x + 7\). The Pythagorean theorem relates the sides of a right triangle: the square of the hypotenuse is equal to the sum of the squares of the two legs. Given legs \(a = x\) and \(b = x + 7\), and a hypotenuse \(c = 13\), we apply this theorem.

  1. Solving and Identifying the Equation

The Pythagorean theorem states:
\[a^2 + b^2 = c^2\]
Substituting our variables into this formula, we get:
\[x^2 + (x + 7)^2 = 13^2\]
Next, we calculate the square of the hypotenuse:
\[13^2 = 13 \times 13 = 169\]
Thus, the equation simplifies to:
\[x^2 + (x + 7)^2 = 169\]
Comparing this result to the given choices, we can see that it matches the final option.

Examples

This scenario is a classic application of the Pythagorean theorem, which is essential in architecture, construction, and landscaping. For example, when building a deck or a garden structure with right-angled corners, contractors use this theorem to ensure the frame is perfectly square. By measuring the two legs of a triangle, they can calculate the required length for the diagonal support to ensure structural stability. It is a fundamental tool for precision in any project involving right triangles.

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