Question

ciera is creating a logo. the figure on the grid shows the background. determine the ratio of the area of the inner square to the area of the outer square. the area of the inner square is square units. the area of the outer square is square units. the ratio of the area of the inner square to the area of the outer square is

Explanation:

Step1: Calculate the area of the outer square

The side length of the outer square is \(8\) units (from \(x = 0\) to \(x = 8\) or \(y=2\) to \(y = 10\)). Using the formula for the area of a square \(A = s^{2}\), where \(s\) is the side - length. So, \(A_{outer}=8\times8 = 64\) square units.

Step2: Calculate the area of the inner square

We can use the method of subtracting the areas of the four right - angled triangles from the area of the outer square. Each right - angled triangle has legs of length \(2\) and \(6\). The area of a right - angled triangle is \(A_{\triangle}=\frac{1}{2}\times a\times b\), where \(a = 2\) and \(b = 6\). The area of one triangle is \(\frac{1}{2}\times2\times6=6\) square units. The area of four triangles is \(4\times6 = 24\) square units. Then \(A_{inner}=A_{outer}-4\times A_{\triangle}=64 - 24=40\) square units.

Step3: Calculate the ratio

The ratio of the area of the inner square to the area of the outer square is \(\frac{A_{inner}}{A_{outer}}=\frac{40}{64}=\frac{5}{8}\)

Answer:

The area of the inner square is \(40\) square units. The area of the outer square is \(64\) square units. The ratio of the area of the inner square to the area of the outer square is \(\frac{5}{8}\)