Area of Rotated Square T with S=27 in Nested Squares

Answer

The side length of square $S$ is the difference between the large square and small square side lengths, such that $s = b - a = 27$. By analyzing the geometry of the second diagram, the diagonal of square $T$ is found to be equal to $b - a$, which leads to an area for square $T$ of $364.5$.

Explanation

In the provided image, we see two configurations where a small square of side $a$ is nested in the bottom-left corner of a large square of side $b$. In the first configuration, square $S$ fills the top-right corner with sides parallel to the container. In the second, square $T$ is rotated $45^\circ$, touching the inner square's corner and the outer square's boundaries.

1. Determine the relation for Square $S$ In the left diagram, square $S$ is positioned such that its side length $s$ plus the side length $a$ of the small square must equal the side length $b$ of the large square. $s = b - a$ This equation states that the side of square $S$ is simply the remaining horizontal or vertical gap between the inner and outer squares. Given $s = 27$, we have: $b - a = 27$ This establishes the fixed difference between the dimensions of the two primary squares.

2. Analyze the geometry of Square $T$ ⚠️ This step is required on exams: accurately relating the diagonal of a rotated square to the boundary dimensions. In the right diagram, the diagonals of square $T$ are parallel to the sides of the outer square. Let the side length of square $T$ be $x$. The length of its diagonal $d$ is given by: $d = x\sqrt{2}$ The diagonal represents the distance from the vertex of $T$ to its opposite vertex across a straight line. Looking at the horizontal alignment, square $T$ starts at the edge of the inner square (position $a$) and ends at the edge of the outer square (position $b$). Therefore, the horizontal diagonal of $T$ must span that distance: $d = b - a$ This expresses that the full width of the rotated square fits exactly in the gap between the two squares.

3. Substitute the known values Since we know $b - a = 27$ from the first square, we can set the diagonal $d$ of square $T$ equal to $27$: $x\sqrt{2} = 27$ This equation allows us to find the side length $x$ of square $T$ in terms of the known gap.

4. Calculate the Area of Square $T$ The area $A$ of a square can be calculated using its side length ($A = x^2$) or its diagonal ($A = \frac{d^2}{2}$): $A = \frac{d^2}{2}$ This formula derives from the fact that a square's area is half the product of its diagonals. Substitute $d = 27$: $A = \frac{27^2}{2}$ $A = \frac{729}{2}$ $A = 364.5$ The resulting value represents the total surface area of the rotated square $T$.

Final Answer

The area of square $T$ is calculated as: $\boxed{364.5}$

Common Mistakes

• Confusing Sides and Diagonals: A common error is assuming the side length of square $T$ is $b-a$. In the rotated orientation, it is the diagonal that spans the distance $b-a$, not the side.
• Calculation Error: Squaring $27$ incorrectly (it is $729$, not $629$ or $829$) can lead to an incorrect final area. Always verify your squares when calculators are not permitted.