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The area of an isosceles trapezium is 90 cm2 and the height is 5/9 th of the sum of its parallel sides. If the ratio of the length of the parallel sides is 4 : 5, then the length of a diagonal (in cm) is
- a)2√137
- b)√181
- c)9√5
- d)18√3
Correct answer is option 'B'. Can you explain this answer?
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The area of an isosceles trapezium is given by the formula:
Area = (1/2) * (sum of parallel sides) * height
Let the lengths of the parallel sides be 4x and 5x, where x is a common factor.
Given that the area is 90 cm^2 and the height is (5/9) times the sum of the parallel sides, we can write the equation:
90 = (1/2) * (4x + 5x) * (5/9) * (4x + 5x)
Simplifying this equation, we get:
90 = (9/2) * 9x^2
Dividing both sides by (9/2), we get:
10 = 9x^2
Simplifying further, we get:
x^2 = 10/9
Taking the square root of both sides, we get:
x = √(10/9)
Since the length of the diagonal is given by the formula:
Diagonal = √((difference of parallel sides)^2 + height^2)
Let's substitute the values into this formula:
Diagonal = √((5x - 4x)^2 + (5/9)(4x + 5x)^2)
Diagonal = √(x^2 + (25/9)(9x)^2)
Diagonal = √(x^2 + (25/9)(81x^2))
Diagonal = √(x^2 + 625x^2)
Diagonal = √(626x^2)
Diagonal = x√626
Substituting the value of x we found earlier, we get:
Diagonal = (√(10/9)) * √626
Hence, the length of the diagonal is (√(10/9)) * √626 cm.
Area = (1/2) * (sum of parallel sides) * height
Let the lengths of the parallel sides be 4x and 5x, where x is a common factor.
Given that the area is 90 cm^2 and the height is (5/9) times the sum of the parallel sides, we can write the equation:
90 = (1/2) * (4x + 5x) * (5/9) * (4x + 5x)
Simplifying this equation, we get:
90 = (9/2) * 9x^2
Dividing both sides by (9/2), we get:
10 = 9x^2
Simplifying further, we get:
x^2 = 10/9
Taking the square root of both sides, we get:
x = √(10/9)
Since the length of the diagonal is given by the formula:
Diagonal = √((difference of parallel sides)^2 + height^2)
Let's substitute the values into this formula:
Diagonal = √((5x - 4x)^2 + (5/9)(4x + 5x)^2)
Diagonal = √(x^2 + (25/9)(9x)^2)
Diagonal = √(x^2 + (25/9)(81x^2))
Diagonal = √(x^2 + 625x^2)
Diagonal = √(626x^2)
Diagonal = x√626
Substituting the value of x we found earlier, we get:
Diagonal = (√(10/9)) * √626
Hence, the length of the diagonal is (√(10/9)) * √626 cm.
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The area of an isosceles trapezium is 90 cm2and the height is 5/9 th of the sum of its parallel sides. If the ratio of the length of the parallel sides is 4 : 5, then the length of a diagonal (in cm) isa)2√137b)√181c)9√5d)18√3Correct answer is option 'B'. Can you explain this answer?
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