Question

an architect designs a diagonal path across a rectangular patio. the path is 29 meters long. the width of the patio is x meters, and the length of the path is 5 meters more than the width.
image of a rectangle with a diagonal labeled 29 m, width labeled x m, length labeled (x + 5) m, and text not drawn to scale
which equation can be used to find the dimensions of the patio?
○ 0.5(x)(x + 5) = 29
○ 0.5(x)(x + 5) = 841
○ x² + (x + 5)² = 29
○ x² + (x + 5)² = 841

Explanation:

Step1: Identify the triangle type

The diagonal of a rectangle forms a right - triangle with the length and width of the rectangle. So we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\).

Step2: Define the sides

In this case, the width of the patio is \(x\) meters (one leg of the right - triangle), the length of the patio is \((x + 5)\) meters (the other leg of the right - triangle), and the diagonal (hypotenuse) is 29 meters.

Step3: Apply the Pythagorean theorem

According to the Pythagorean theorem, \(x^{2}+(x + 5)^{2}=29^{2}\). Since \(29^{2}=841\), the equation becomes \(x^{2}+(x + 5)^{2}=841\).

Answer:

\(x^{2}+(x + 5)^{2}=841\) (the last option: \(x^{2}+(x + 5)^{2}=841\))