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 11 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 side - length of square

In a square, if the diagonal \(d = 11\) inches, and using the Pythagorean theorem \(d^{2}=s^{2}+s^{2}\) (where \(s\) is the side - length of the square), so \(11^{2}=2s^{2}\), then \(s^{2}=\frac{11^{2}}{2}=\frac{121}{2}\), and \(s=\frac{11}{\sqrt{2}}\approx\frac{11}{1.414}\approx7.78\) inches.

Step2: Calculate perimeter of square

The perimeter \(P\) of a square with side - length \(s\) is \(P = 4s\). Substituting \(s\approx7.78\) inches, we get \(P = 4\times7.78 = 31.12\) inches.

Step3: Calculate total length of diagonals

The total length of the two diagonals is \(2d\). Since \(d = 11\) inches, the total length of the two diagonals is \(2\times11=22\) inches.

Step4: Calculate total length of iron edging

The total length \(L\) of iron edging is the sum of the perimeter of the square and the total length of the two diagonals. So \(L=31.12 + 22=53.12\approx54.0\) inches.

Answer:

54.0 inches