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The area of a rectangle is equal to the area of a square whose diagonal is 12√2 metre. The difference between the length and the breadth of the rectangle is 7 metre. What is the perimeter of rectangle ? (in metre).
- a)68 metre
- b)50 metre
- c)62 metre
- d)64 metre
- e)None of the Above
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The area of a rectangle is equal to the area of a square whose diagona...
d = a√2
12√2 = a√2
a = 12
l * b = a² = (12²) = 144
l – b = 7 ; l = b + 7
(b + 7)*(b) = 144
b² + 7b – 144 = 0
b = 9; l = 16
2(l + b) = 2(16 + 9) = 50m
12√2 = a√2
a = 12
l * b = a² = (12²) = 144
l – b = 7 ; l = b + 7
(b + 7)*(b) = 144
b² + 7b – 144 = 0
b = 9; l = 16
2(l + b) = 2(16 + 9) = 50m
Most Upvoted Answer
The area of a rectangle is equal to the area of a square whose diagona...
Understanding the Problem
To find the perimeter of the rectangle, we first need to determine its dimensions based on the information given.
Step 1: Find the Area of the Square
- The diagonal of the square is given as 12√2 metres.
- Using the relationship between the side (s) and the diagonal (d) of a square: d = s√2.
- Thus, s = d/√2 = (12√2)/√2 = 12 metres.
- The area of the square = s² = 12² = 144 square metres.
Step 2: Set Up the Rectangle's Dimensions
- Let the length of the rectangle be L and the breadth be B.
- We know that the area of the rectangle = L × B = 144 square metres.
- We're also given that the difference between the length and the breadth is 7 metres: L - B = 7.
Step 3: Solve the Equations
- From L - B = 7, we can express L as: L = B + 7.
- Substitute L in the area equation: (B + 7) × B = 144.
- Expanding this gives: B² + 7B - 144 = 0.
Step 4: Factor the Quadratic Equation
- We need to factor B² + 7B - 144.
- The factors are (B + 16)(B - 9) = 0.
- Thus, B = 9 (since breadth cannot be negative) and L = B + 7 = 9 + 7 = 16 metres.
Step 5: Calculate the Perimeter
- The perimeter (P) of the rectangle is given by the formula: P = 2(L + B).
- Substitute L and B: P = 2(16 + 9) = 2 × 25 = 50 metres.
Conclusion
- The perimeter of the rectangle is 50 metres, which corresponds to option 'B'.
To find the perimeter of the rectangle, we first need to determine its dimensions based on the information given.
Step 1: Find the Area of the Square
- The diagonal of the square is given as 12√2 metres.
- Using the relationship between the side (s) and the diagonal (d) of a square: d = s√2.
- Thus, s = d/√2 = (12√2)/√2 = 12 metres.
- The area of the square = s² = 12² = 144 square metres.
Step 2: Set Up the Rectangle's Dimensions
- Let the length of the rectangle be L and the breadth be B.
- We know that the area of the rectangle = L × B = 144 square metres.
- We're also given that the difference between the length and the breadth is 7 metres: L - B = 7.
Step 3: Solve the Equations
- From L - B = 7, we can express L as: L = B + 7.
- Substitute L in the area equation: (B + 7) × B = 144.
- Expanding this gives: B² + 7B - 144 = 0.
Step 4: Factor the Quadratic Equation
- We need to factor B² + 7B - 144.
- The factors are (B + 16)(B - 9) = 0.
- Thus, B = 9 (since breadth cannot be negative) and L = B + 7 = 9 + 7 = 16 metres.
Step 5: Calculate the Perimeter
- The perimeter (P) of the rectangle is given by the formula: P = 2(L + B).
- Substitute L and B: P = 2(16 + 9) = 2 × 25 = 50 metres.
Conclusion
- The perimeter of the rectangle is 50 metres, which corresponds to option 'B'.
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