Question

a windowpane is 7 inches tall and the diagonal distance from one corner to the other is 9 inches. how wide is the windowpane? if necessary, round to the nearest tenth. inches

Explanation:

Step1: Identify the problem type

This is a right triangle problem where the windowpane is a rectangle, so the height, width, and diagonal form a right triangle. We can use the Pythagorean theorem, which is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (diagonal), and \(a\) and \(b\) are the other two sides (height and width). Let the height \(a = 7\) inches, the diagonal \(c=9\) inches, and the width be \(b\).

Step2: Rearrange the Pythagorean theorem to solve for \(b\)

We know that \(a^{2}+b^{2}=c^{2}\), so we can rearrange it to \(b^{2}=c^{2}-a^{2}\). Substitute \(a = 7\) and \(c = 9\) into the formula: \(b^{2}=9^{2}-7^{2}\). Calculate \(9^{2}=81\) and \(7^{2}=49\), so \(b^{2}=81 - 49=32\).

Step3: Solve for \(b\)

To find \(b\), we take the square root of both sides: \(b=\sqrt{32}\). Simplify \(\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\approx4\times1.414 = 5.656\). Rounding to the nearest tenth, we get \(b\approx5.7\).

Answer:

\(5.7\)