The stained-glass window below shows bilateral symmetry. The two overlapping squares are congruent. What is the area of the window?
The correct answer and explanation is:
To determine the area of the stained-glass window, we need to use the key information given:
- The window shows bilateral symmetry.
- The two overlapping squares are congruent.
Step-by-Step Explanation
Assume each square has a side length of 10 units (this would typically be provided or measured from an image, but since we don’t have specific numbers in the problem, let’s use a general method and then assign values for clarity).
When two congruent squares overlap at a 45° angle, they form an eight-pointed star (like a starburst). Despite the overlap, to find the total area of the window, we don’t subtract the overlapping region, because we are asked for the area of the entire window, including both full squares.
Since the two squares are congruent, the total area is simply: Area of window=Area of first square+Area of second square\text{Area of window} = \text{Area of first square} + \text{Area of second square}
But because they overlap, we must subtract the overlapping area once to avoid double-counting.
Let’s now assume:
- Each square has a side length of 10 units.
- So, the area of one square is:
10×10=100 square units10 \times 10 = 100 \text{ square units}
- Area of two squares:
2×100=200 square units2 \times 100 = 200 \text{ square units}
If the squares are rotated and perfectly overlap in part, we must subtract the overlapping region. In most such symmetric star windows, the overlapping region forms a regular octagon or similar shape, and its area can be computed geometrically. However, if the problem doesn’t specify dimensions or angles, and only wants the area covered by the full figure, and if the full window is framed as the union of both squares (not just the overlapped shape), then we must consider the bounding figure.
Final Answer:
If the squares are fully overlapping in a star pattern and the window is the entire region covered by both, then: 200 square units\boxed{200 \text{ square units}}
(assuming side length of 10 units; if another side length is provided, use s2+s2=2s2s^2 + s^2 = 2s^2 for total area).