Question

Could someone help me please

Answer 1

To find the ratio of the areas, we are obviously going to first need to find the areas of the inner and outer squares. The outer square has a side length of $a$ because we are $a$ units away from 0 on the x-axis. Thus, the area of the outer square is $a^2$.

We can see that a right triangle is formed by part of the x-axis, part of the y-axis, and the side of the inner square. Thus, we can find the length of the inner square through the Pythagorean Theorem, which is $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of the right triangle and $c$ is the hypotenuse. The lengths of the legs of the right triangle in the picture are $(a - b)$ and $b$. We can use the Pythagorean Theorem to find the other side length.

$(a - b)^2 + b^2 = c^2$

$c = \sqrt{(a - b)^2 + b^2}$

We have now found that the side length of the inner square is $\sqrt{(a -b)^2 + b^2}$. Thus, the area of the inner square is $(a - b)^2 + b^2$.

Using the two areas we just found, we can say that the ratio of the area of the inner square to the area of the outer square is $\dfrac{(a - b)^2 + b^2}{a^2}$, or choice A.