A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. If the ratios $9:\text{S}$ and $\text{S}:\text{T}$ are both equal to 3 and $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares, what is the total shaded area?

Step-by-step explanation:

I attached an image to aid the understanding of the question.

Looking at the image, we see that the 8 shaded parts are congruent, as affirmed in the question as well. And we are told that T = 3, this implies that the area of the square with T as it's side is 9ft². Since all the 8 squares are congruenrt, it means each square has its area to be 9. Therefore, the total area of the 8 shaded squares will be:

It remains the area of the shaded square with side S.

From the question, we have the following ratio:

I did not add ± because length is always positive, so the case of negative is eliminated.

Now the areas of S is

Therefore, the total area of the shaded squares is