Question

a square is inscribed in another square, as shown in the figure below. what percentage of the larger squares area is the smaller squares area? a 43% b 50%

Explanation:

Step1: Find side - length of larger square

The side - length of the larger square is \(3 + 2=5\). So the area of the larger square \(A_{1}=5\times5 = 25\).

Step2: Use the Pythagorean theorem for right - triangles

The right - triangles formed at the corners of the larger square have legs of lengths 2 and 3. By the Pythagorean theorem, the side - length of the smaller square \(s=\sqrt{2^{2}+3^{2}}=\sqrt{4 + 9}=\sqrt{13}\). Then the area of the smaller square \(A_{2}=13\).

Step3: Calculate the percentage

The percentage of the smaller square's area to the larger square's area is \(\frac{A_{2}}{A_{1}}\times100=\frac{13}{25}\times100 = 52\%\). But if we assume there is a mistake in the above and consider the four right - triangles. The sum of the areas of the four right - triangles with legs \(a = 2\) and \(b = 3\) is \(4\times\frac{1}{2}\times2\times3=12\). The area of the larger square \(A_{l}=(2 + 3)^{2}=25\). The area of the smaller square \(A_{s}=25-12 = 13\). The percentage is \(\frac{13}{25}\times100 = 52\%\).

Answer:

52%