Question

given the figure shown below, determine the length of a.

Explanation:

Step1: Find the side length of the square with area \(49\mathrm{in}^2\)

Let the side length of the square with area \(49\mathrm{in}^2\) be \(b\). Using the formula for the area of a square \(A = b^2\), we have \(b^2=49\), so \(b = 7\mathrm{in}\) (since \(b>0\)).

Step2: Apply the Pythagorean theorem

Let the side length of the square with area \(113\mathrm{in}^2\) be \(c\). Using the formula for the area of a square \(A = c^2\), we have \(c^2 = 113\).
By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Substitute \(b = 7\) and \(c^{2}=113\) into the equation: \(a^{2}+49 = 113\).

Step3: Solve for \(a\)

Subtract \(49\) from both sides of the equation \(a^{2}+49 = 113\):
\(a^{2}=113 - 49\)
\(a^{2}=64\)
Take the square - root of both sides (\(a>0\)), so \(a=\sqrt{64}=8\mathrm{in}\)

Answer:

\(8\mathrm{in}\)