Question

a square stained glass window is divided into four congruent triangular sections by iron edging to represent the seasons of the year. each diagonal of the square window measures 9 inches. what is the approximate total length of iron edging needed to create the square frame and the two diagonals? 43.5 inches 50.9 inches 54.0 inches 61.5 inches

Explanation:

Step1: Find the side length of the square

Let the side length of the square be \(a\). Using the Pythagorean theorem for a square (\(a^{2}+a^{2}=d^{2}\), where \(d\) is the diagonal). Given \(d = 9\) inches, we have \(2a^{2}=81\), so \(a^{2}=\frac{81}{2}\), and \(a=\frac{9}{\sqrt{2}}=\frac{9\sqrt{2}}{2}\approx6.36\) inches.

Step2: Calculate the perimeter of the square

The perimeter \(P\) of a square is \(4a\). So \(P = 4\times\frac{9\sqrt{2}}{2}=18\sqrt{2}\approx25.46\) inches.

Step3: Calculate the total length of the diagonals

The total length of the two diagonals is \(2d\). Since \(d = 9\) inches, the length of the two diagonals is \(2\times9 = 18\) inches.

Step4: Calculate the total length of the iron - edging

The total length \(L\) of the iron - edging is the sum of the perimeter of the square and the lengths of the two diagonals. So \(L\approx25.46+18 = 43.46\approx43.5\) inches.

Answer:

43.5 inches