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The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, is
- a)24
- b)27
- c)21
- d)18
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The sum of the perimeters of an equilateral triangle and a rectangle i...
Let the breadth of the rectangle be b.
Length of the rectangle 3b
Let a be the side of the equilateral triangle.
Given,
Given,
2(4b) + 3a = 90
8(a2 / 4)+ 3a - 90 = 0
⇒ 2a2 + 3a - 90 = 0
⇒ (a - 6)(2a +15) = 0
⇒ a = 6
⇒ b = 9
⇒ 3b = 27
Length of the rectangle 3b
Let a be the side of the equilateral triangle.
Given,
Given,
2(4b) + 3a = 90
8(a2 / 4)+ 3a - 90 = 0
⇒ 2a2 + 3a - 90 = 0
⇒ (a - 6)(2a +15) = 0
⇒ a = 6
⇒ b = 9
⇒ 3b = 27
Most Upvoted Answer
The sum of the perimeters of an equilateral triangle and a rectangle i...
Let's assume that the equilateral triangle has side length 'a' cm and the sides of the rectangle are 'x' cm and '3x' cm.
Perimeter of the equilateral triangle:
The perimeter of an equilateral triangle is equal to the sum of the lengths of its three sides. Since all sides of an equilateral triangle are equal, the perimeter is 3 times the length of any one side.
Perimeter of the equilateral triangle = 3a cm
Perimeter of the rectangle:
The perimeter of a rectangle is equal to twice the sum of its length and width. Since the sides of the rectangle are 'x' cm and '3x' cm, the perimeter is:
Perimeter of the rectangle = 2(x + 3x) = 2(4x) = 8x cm
Given that the sum of the perimeters of the equilateral triangle and the rectangle is 90 cm, we can write the equation:
3a + 8x = 90
Area of the equilateral triangle:
The area of an equilateral triangle can be calculated using the formula: Area = (√3/4) * side^2
Area of the equilateral triangle = (√3/4) * a^2
Area of the rectangle:
The area of a rectangle is equal to the product of its length and width. Since the sides of the rectangle are 'x' cm and '3x' cm, the area is:
Area of the rectangle = x * 3x = 3x^2
Given that the area of the rectangle is the square of the area of the equilateral triangle, we can write the equation:
3x^2 = (√3/4) * a^2
Simplifying, we get:
x^2 = (√3/12) * a^2
Solving the two equations:
We have two equations:
3a + 8x = 90
x^2 = (√3/12) * a^2
We can substitute the value of a^2 from the second equation into the first equation:
3a + 8x = 90
3a + 8(√3/12) * x^2 = 90
3a + (√3/3) * x^2 = 90
3a + (√3/3) * 3x^2 = 90
3a + (√3/3) * 3(√3/12) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * (a^2) = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3
Perimeter of the equilateral triangle:
The perimeter of an equilateral triangle is equal to the sum of the lengths of its three sides. Since all sides of an equilateral triangle are equal, the perimeter is 3 times the length of any one side.
Perimeter of the equilateral triangle = 3a cm
Perimeter of the rectangle:
The perimeter of a rectangle is equal to twice the sum of its length and width. Since the sides of the rectangle are 'x' cm and '3x' cm, the perimeter is:
Perimeter of the rectangle = 2(x + 3x) = 2(4x) = 8x cm
Given that the sum of the perimeters of the equilateral triangle and the rectangle is 90 cm, we can write the equation:
3a + 8x = 90
Area of the equilateral triangle:
The area of an equilateral triangle can be calculated using the formula: Area = (√3/4) * side^2
Area of the equilateral triangle = (√3/4) * a^2
Area of the rectangle:
The area of a rectangle is equal to the product of its length and width. Since the sides of the rectangle are 'x' cm and '3x' cm, the area is:
Area of the rectangle = x * 3x = 3x^2
Given that the area of the rectangle is the square of the area of the equilateral triangle, we can write the equation:
3x^2 = (√3/4) * a^2
Simplifying, we get:
x^2 = (√3/12) * a^2
Solving the two equations:
We have two equations:
3a + 8x = 90
x^2 = (√3/12) * a^2
We can substitute the value of a^2 from the second equation into the first equation:
3a + 8x = 90
3a + 8(√3/12) * x^2 = 90
3a + (√3/3) * x^2 = 90
3a + (√3/3) * 3x^2 = 90
3a + (√3/3) * 3(√3/12) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * (a^2) = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3a + (√3/3) * (√3/4) * a^2 = 90
3
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The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, isa)24b)27c)21d)18Correct answer is option 'B'. Can you explain this answer?
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The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, isa)24b)27c)21d)18Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, isa)24b)27c)21d)18Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, isa)24b)27c)21d)18Correct answer is option 'B'. Can you explain this answer?.
The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, isa)24b)27c)21d)18Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, isa)24b)27c)21d)18Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. Thearea, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy therelationship R = T2 . If the sides of the rectangle are in the ratio 1: 3, then the length, in cm,of the longer side of the rectangle, isa)24b)27c)21d)18Correct answer is option 'B'. Can you explain this answer?.
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