The length and breadth of a rectangle are in the ratio $9:7$. If the perimeter of the rectangle is $128 ext{ cm}$ and the side of a square is $6 ext{ cm}$ less than the length of the rectangle, what is the area of the square?

Question

The length and breadth of a rectangle are in the ratio $9:7$. If the perimeter of the rectangle is $128	ext{ cm}$ and the side of a square is $6	ext{ cm}$ less than the length of the rectangle, what is the area of the square?

Accepted Answer

**Solution:**

1. Let the length and breadth of the rectangle be $9x$ and $7x$ respectively.

2. The perimeter of the rectangle is given by:
   $$2(	ext{length} + 	ext{breadth}) = 128$$
   Substituting values:
   $$2(9x + 7x) = 128$$
   $$2(16x) = 128$$
   $$32x = 128$$
   $$x = \frac{128}{32} = 4$$

3. Therefore, the length of the rectangle is:
   $$9x = 9 	imes 4 = 36 	ext{ cm}$$

4. The side of the square is $6 	ext{ cm}$ less than the length of the rectangle:
   $$	ext{side of square} = 36 - 6 = 30 	ext{ cm}$$

5. The area of the square is:
   $$	ext{side}^2 = 30^2 = 900 	ext{ cm}^2$$

**Answer:** $900 	ext{ cm}^2$

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