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The diagonal of a rectangle is 17 cm and its area is 120 cm2. The perimeter of the rectangle is ____
- a)23 cm
- b)32 cm
- c)30 cm
- d)46 cm
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The diagonal of a rectangle is 17 cm and its area is 120 cm2. The peri...
Given:
The diagonal of a rectangle = 17 cm
The area of the rectangle = 120 cm2
Formula used:
The diagonal of a rectangle = √(Length2 + Width2)
The area of a rectangle = Length × Width
The perimeter of a rectangle = 2 (Length + Width)
Calculation:
Let, the length of the rectangle = L
The width of the rectangle = W
Then, L × W = 120 .....(1)
Also, √(L2 + W2) = 17
⇒ (L2 + W2) = 172 [Squaring on both sides]
⇒ (L2 + W2) = 289
⇒ (L + W)2 - 2 × L × W = 289 [By using a2 + b2 = (a + b)2 - 2ab formula]
⇒ (L + W)2 - 2 × 120 = 289 [From the equation (1), L × W = 120]
⇒ (L + W)2 - 240 = 289
⇒ (L + W)2 = 289 + 240
⇒ (L + W)2 = 529
⇒ (L + W) = √529 [Taking square root of both sides]
⇒ (L + W) = 23 .....(2)
Now, The perimeter of the rectangle = 2 (L + W)
= 2 × 23 [∵ (L + W) = 23]
= 46 cm
∴ The perimeter of the rectangle is 46 cm.
The diagonal of a rectangle = 17 cm
The area of the rectangle = 120 cm2
Formula used:
The diagonal of a rectangle = √(Length2 + Width2)
The area of a rectangle = Length × Width
The perimeter of a rectangle = 2 (Length + Width)
Calculation:
Let, the length of the rectangle = L
The width of the rectangle = W
Then, L × W = 120 .....(1)
Also, √(L2 + W2) = 17
⇒ (L2 + W2) = 172 [Squaring on both sides]
⇒ (L2 + W2) = 289
⇒ (L + W)2 - 2 × L × W = 289 [By using a2 + b2 = (a + b)2 - 2ab formula]
⇒ (L + W)2 - 2 × 120 = 289 [From the equation (1), L × W = 120]
⇒ (L + W)2 - 240 = 289
⇒ (L + W)2 = 289 + 240
⇒ (L + W)2 = 529
⇒ (L + W) = √529 [Taking square root of both sides]
⇒ (L + W) = 23 .....(2)
Now, The perimeter of the rectangle = 2 (L + W)
= 2 × 23 [∵ (L + W) = 23]
= 46 cm
∴ The perimeter of the rectangle is 46 cm.
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Community Answer
The diagonal of a rectangle is 17 cm and its area is 120 cm2. The peri...
Given information:
- Diagonal of the rectangle = 17 cm
- Area of the rectangle = 120 cm²
To find:
- Perimeter of the rectangle
Solution:
1. Finding the sides of the rectangle:
- Let the length of the rectangle be 'l' cm.
- Let the breadth of the rectangle be 'b' cm.
2. Using the Pythagorean theorem:
- In a rectangle, the diagonal, length, and breadth form a right-angled triangle.
- According to the Pythagorean theorem, the square of the length of the diagonal is equal to the sum of the squares of the length and breadth.
- Therefore, we have the equation: l² + b² = 17²
3. Finding the area:
- The area of a rectangle is given by the product of its length and breadth.
- We are given that the area of the rectangle is 120 cm².
- Therefore, we have the equation: l * b = 120
4. Solving the equations:
- We have two equations:
- l² + b² = 17²
- l * b = 120
- We can solve these equations simultaneously to find the values of length and breadth.
5. Simplifying the equations:
- Rearranging the area equation, we get: l = 120/b
- Substituting this value of l in the diagonal equation, we get: (120/b)² + b² = 17²
6. Solving the quadratic equation:
- Expanding the equation and simplifying, we get: 14400/b² + b² = 289
- Multiplying through by b², we get: 14400 + b⁴ = 289b²
- Rearranging the terms, we get: b⁴ - 289b² + 14400 = 0
7. Solving the quadratic equation:
- This quadratic equation can be factored as: (b² - 160)(b² - 90) = 0
- Solving the individual equations, we get: b = ±√160 or b = ±√90
8. Selecting the valid solution:
- Since the breadth cannot be negative, we take the positive square root.
- Therefore, b = √160 or b = √90
9. Calculating the length:
- Using the area equation, we have: l = 120/b
- Substituting the value of b, we get: l = 120/√160 or l = 120/√90
10. Simplifying the length:
- Rationalizing the denominators, we get: l = (120√160)/160 or l = (120√90)/90
- Simplifying further, we get: l = 3√10 cm or l = 4√10/3 cm
11. Calculating the perimeter:
- The perimeter of a rectangle is given by the sum of all its sides.
- Therefore, the perimeter of the rectangle = 2l + 2b
12. Substituting the values:
- Substituting the values of length and breadth, we get:
- Perimeter = 2(3√10) + 2(√160) or Perimeter = 2(4√10/3) +
- Diagonal of the rectangle = 17 cm
- Area of the rectangle = 120 cm²
To find:
- Perimeter of the rectangle
Solution:
1. Finding the sides of the rectangle:
- Let the length of the rectangle be 'l' cm.
- Let the breadth of the rectangle be 'b' cm.
2. Using the Pythagorean theorem:
- In a rectangle, the diagonal, length, and breadth form a right-angled triangle.
- According to the Pythagorean theorem, the square of the length of the diagonal is equal to the sum of the squares of the length and breadth.
- Therefore, we have the equation: l² + b² = 17²
3. Finding the area:
- The area of a rectangle is given by the product of its length and breadth.
- We are given that the area of the rectangle is 120 cm².
- Therefore, we have the equation: l * b = 120
4. Solving the equations:
- We have two equations:
- l² + b² = 17²
- l * b = 120
- We can solve these equations simultaneously to find the values of length and breadth.
5. Simplifying the equations:
- Rearranging the area equation, we get: l = 120/b
- Substituting this value of l in the diagonal equation, we get: (120/b)² + b² = 17²
6. Solving the quadratic equation:
- Expanding the equation and simplifying, we get: 14400/b² + b² = 289
- Multiplying through by b², we get: 14400 + b⁴ = 289b²
- Rearranging the terms, we get: b⁴ - 289b² + 14400 = 0
7. Solving the quadratic equation:
- This quadratic equation can be factored as: (b² - 160)(b² - 90) = 0
- Solving the individual equations, we get: b = ±√160 or b = ±√90
8. Selecting the valid solution:
- Since the breadth cannot be negative, we take the positive square root.
- Therefore, b = √160 or b = √90
9. Calculating the length:
- Using the area equation, we have: l = 120/b
- Substituting the value of b, we get: l = 120/√160 or l = 120/√90
10. Simplifying the length:
- Rationalizing the denominators, we get: l = (120√160)/160 or l = (120√90)/90
- Simplifying further, we get: l = 3√10 cm or l = 4√10/3 cm
11. Calculating the perimeter:
- The perimeter of a rectangle is given by the sum of all its sides.
- Therefore, the perimeter of the rectangle = 2l + 2b
12. Substituting the values:
- Substituting the values of length and breadth, we get:
- Perimeter = 2(3√10) + 2(√160) or Perimeter = 2(4√10/3) +
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