a square stained glass window is divided into four congruent triangular sections by iron edging to represent…

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

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

Answer

Explanation:

Step1: Find side - length of square

In a square, if the diagonal length is $d$, and the side - length is $s$, then by the Pythagorean theorem $d^{2}=s^{2}+s^{2}=2s^{2}$. Given $d = 9$ inches, we have $9^{2}=2s^{2}$, so $s^{2}=\frac{81}{2}$, and $s=\frac{9}{\sqrt{2}}=\frac{9\sqrt{2}}{2}\approx\frac{9\times1.414}{2}=6.363$ inches.

Step2: Calculate perimeter of square

The perimeter $P$ of a square with side - length $s$ is $P = 4s$. Substituting $s=\frac{9\sqrt{2}}{2}$ inches, we get $P = 4\times\frac{9\sqrt{2}}{2}=18\sqrt{2}\approx18\times1.414 = 25.452$ inches.

Step3: Calculate total length of diagonals

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

Step4: Calculate total length of iron edging

The total length $L$ of the iron edging is the sum of the perimeter of the square and the total length of the two diagonals. So $L=18\sqrt{2}+18\approx25.452 + 18=43.452\approx43.5$ inches.

Answer:

43.5 inches