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The ratio of length of each equal side and the third side of an isosceles triangle is 3 : 4. If the area of the triangle is 18√5 sq units, the third side is
- a)8√2 units
- b)12 units
- c)16 units
- d)5√10 units
Correct answer is option 'B'. Can you explain this answer?
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Let the length of each equal side be 3x and the length of the third side be 4x.
The area of an isosceles triangle can be calculated using the formula:
Area = (1/2) * base * height
In this case, the base is 4x and the height can be found using the Pythagorean theorem. Let's call the height h.
Using the Pythagorean theorem, we can write the equation:
(3x)^2 = h^2 + (4x/2)^2
9x^2 = h^2 + 4x^2
5x^2 = h^2
Substituting into the area formula:
Area = (1/2) * 4x * sqrt(5x^2)
Area = 2x * x * sqrt(5)
Area = 2x^2 * sqrt(5)
Given that the area is 18, we can set up the equation:
2x^2 * sqrt(5) = 18
x^2 * sqrt(5) = 9
x^2 = 9 / sqrt(5)
x = sqrt(9 / sqrt(5))
x = sqrt(9) / sqrt(sqrt(5))
x = 3 / sqrt(2)
Therefore, the length of each equal side is 3 * (3 / sqrt(2)) = 9 / sqrt(2) and the length of the third side is 4 * (3 / sqrt(2)) = 12 / sqrt(2).
Simplifying the lengths, we get:
Length of each equal side ≈ 6.36
Length of the third side ≈ 8.49
The area of an isosceles triangle can be calculated using the formula:
Area = (1/2) * base * height
In this case, the base is 4x and the height can be found using the Pythagorean theorem. Let's call the height h.
Using the Pythagorean theorem, we can write the equation:
(3x)^2 = h^2 + (4x/2)^2
9x^2 = h^2 + 4x^2
5x^2 = h^2
Substituting into the area formula:
Area = (1/2) * 4x * sqrt(5x^2)
Area = 2x * x * sqrt(5)
Area = 2x^2 * sqrt(5)
Given that the area is 18, we can set up the equation:
2x^2 * sqrt(5) = 18
x^2 * sqrt(5) = 9
x^2 = 9 / sqrt(5)
x = sqrt(9 / sqrt(5))
x = sqrt(9) / sqrt(sqrt(5))
x = 3 / sqrt(2)
Therefore, the length of each equal side is 3 * (3 / sqrt(2)) = 9 / sqrt(2) and the length of the third side is 4 * (3 / sqrt(2)) = 12 / sqrt(2).
Simplifying the lengths, we get:
Length of each equal side ≈ 6.36
Length of the third side ≈ 8.49
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The ratio of length of each equal side and the third side of an isosceles triangle is 3 : 4. If the area of the triangle is 18√5 sq units, the third side isa)8√2 unitsb)12 unitsc)16 unitsd)5√10 unitsCorrect answer is option 'B'. Can you explain this answer?
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The ratio of length of each equal side and the third side of an isosceles triangle is 3 : 4. If the area of the triangle is 18√5 sq units, the third side isa)8√2 unitsb)12 unitsc)16 unitsd)5√10 unitsCorrect answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about The ratio of length of each equal side and the third side of an isosceles triangle is 3 : 4. If the area of the triangle is 18√5 sq units, the third side isa)8√2 unitsb)12 unitsc)16 unitsd)5√10 unitsCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The ratio of length of each equal side and the third side of an isosceles triangle is 3 : 4. If the area of the triangle is 18√5 sq units, the third side isa)8√2 unitsb)12 unitsc)16 unitsd)5√10 unitsCorrect answer is option 'B'. Can you explain this answer?.
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