7.5. In a large square hall, two square carpets were brought, the side of one carpet being twice the side of the other. When they were placed in opposite corners of the hall, they covered $4 \mathrm{~m}^{2}$ in two layers, and when they were placed in adjacent corners, they covered 14 m². What are the dimensions of the hall?
Answer. $19 \times 19 \mathrm{~m}^{2}$.

Solution. In the first case, the intersection of the carpets is a square with an area of 4 m² (left figure), so the length of the side of this square is 2 m. In the second case, the intersection is a rectangle, one side of which is also 2 m (right figure). Therefore, the other side of this rectangle is

$14: 2=7$ (m), and this is the length of the side of the smaller carpet. Thus, the side of the larger carpet has a length of 14 m. Since the sides of the carpets overlap by 2 m, the length of the side of the hall is $7+14-2=19($ m $)$.
![](https://cdn.mathpix.com/cropped/2024_05_06_d6834231759d063d9b33g-2.jpg?height=356&width=942&top_left_y=2094&top_left_x=520)

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