The areas of a square and a rectangle are equal. If the length of the rectangle is 12 m more than the side of the square and its breadth is 8 m less than the side of the square, wh... The areas of a square and a rectangle are equal. If the length of the rectangle is 12 m more than the side of the square and its breadth is 8 m less than the side of the square, what is the perimeter of the rectangle? (in m)
Understand the Problem
The problem states that a square and a rectangle have equal areas. The length of the rectangle is 12 meters more than the side of the square, and the breadth is 8 meters less than the side of the square. We need to find the perimeter of the rectangle.
Answer
104
Answer for screen readers
104
Steps to Solve
- Define the variables
Let $s$ be the side of the square.
The length of the rectangle, $l$, is $s + 12$.
The breadth of the rectangle, $b$, is $s - 8$.
- Express the equality of areas
Area of square = $s^2$.
Area of rectangle = $l \times b = (s+12)(s-8)$.
Since the areas are equal:
$$s^2 = (s+12)(s-8)$$
- Solve for $s$
Expand the right side:
$$s^2 = s^2 - 8s + 12s - 96$$
$$s^2 = s^2 + 4s - 96$$
Subtract $s^2$ from both sides:
$$0 = 4s - 96$$
Add 96 to both sides:
$$4s = 96$$
Divide by 4:
$$s = \frac{96}{4} = 24$$
- Find the length and breadth of the rectangle
Length $l = s + 12 = 24 + 12 = 36$ meters.
Breadth $b = s - 8 = 24 - 8 = 16$ meters.
- Calculate the perimeter of the rectangle
Perimeter $P = 2(l + b) = 2(36 + 16) = 2(52) = 104$ meters.
104
More Information
The perimeter of the rectangle is 104 meters.
Tips
A common mistake is to incorrectly expand the product $(s+12)(s-8)$
Another common mistake is an arithmetic error during the simplification or solving for $s$.
Finally, one might find the side of the square correctly but then fail to correctly calculate the length and breadth of the rectangle or the perimeter.
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