Question

a rectangular painting has a diagonal measure of 15 inches and an area of 108 square inches. use the formula for the area of a rectangle and the pythagorean theorem to find the length and width of the painting. the length (the longer side) is and the width (the shorter side) is (simplify your answers.)

Explanation:

Step1: Set up equations

Let the length be \( l \) and the width be \( w \).
We know that the area of a rectangle \( A = lw\), so \( lw=108\).
By the Pythagorean theorem, \(l^{2}+w^{2}=15^{2}=225\).

Step2: Use the identity \((l + w)^{2}=l^{2}+w^{2}+2lw\)

Substitute \(l^{2}+w^{2}=225\) and \(lw = 108\) into the identity:
\((l + w)^{2}=225+2\times108\)
\((l + w)^{2}=225 + 216\)
\((l + w)^{2}=441\)
Take the square root of both sides: \(l + w=\sqrt{441}=21\), so \(w = 21 - l\).

Step3: Substitute \(w = 21 - l\) into \(lw=108\)

\(l(21 - l)=108\)
\(21l-l^{2}=108\)
Rearrange to get a quadratic equation: \(l^{2}-21l + 108=0\)
Factor the quadratic: \(l^{2}-21l+108=(l - 9)(l - 12)=0\)
Set each factor equal to zero: \(l-9 = 0\) gives \(l = 9\); \(l - 12=0\) gives \(l=12\).
If \(l = 9\), then \(w=21 - 9=12\); if \(l = 12\), then \(w=21 - 12 = 9\).

Answer:

The length (the longer side) is \(12\) inches and the width (the shorter side) is \(9\) inches.