Question

3 select the correct answer. what is the ratio of the area of the inner square to the area of the outer square? a. $\frac{(a - b)^2 + b^2}{a^2}$ b. $\frac{a^2 - b^2}{a^2}$ c. $\frac{(a - b)^2}{(a + b)^2}$ d. $\frac{(a - b)^2}{(a + b)^2}$

Explanation:

Step1: Find the side - length of the outer square

The side - length of the outer square is \(a\) (from the coordinates \((0,0)\) and \((a,0)\)). So the area of the outer square \(A_{outer}=a^{2}\).

Step2: Find the side - length of the inner square using the distance formula

The distance between two adjacent vertices of the inner square, say \((0,b)\) and \((a - b,0)\), is calculated using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\). Here, \(x_1 = 0,y_1 = b,x_2=a - b,y_2 = 0\). Then \(d=\sqrt{(a - b-0)^{2}+(0 - b)^{2}}=\sqrt{(a - b)^{2}+b^{2}}\). So the area of the inner square \(A_{inner}=(a - b)^{2}+b^{2}=a^{2}-2ab + b^{2}+b^{2}=a^{2}-2ab + 2b^{2}\).

Step3: Calculate the ratio of the areas

The ratio of the area of the inner square to the area of the outer square is \(\frac{A_{inner}}{A_{outer}}=\frac{(a - b)^{2}+b^{2}}{a^{2}}\).

Answer:

A. \(\frac{(a - b)^{2}+b^{2}}{a^{2}}\)