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The length of each side of an equilateral triangle ABC is 3 cm. Let D be a point on BC such that the area of triangle ADC is half the area of triangle ABD. Then the length of AD. in cm, is
- a)√7
- b)√8
- c)√6
- d)√5
Correct answer is option 'A'. Can you explain this answer?
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The length of each side of an equilateral triangle ABC is 3 cm. Let D ...
To solve this problem, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Since triangle ABC is an equilateral triangle, all sides have the same length, which is 3 cm. Therefore, the height of triangle ABC is given by:
Height = (sqrt(3)/2) * side = (sqrt(3)/2) * 3 = (3sqrt(3))/2 cm
Let AD = x cm. Then, the base of triangle ADC is BC - AD = 3 - x cm.
The area of triangle ADC is given as half the area of triangle ABD. Therefore:
(1/2) * (1/2) * (3 - x) * ((3sqrt(3))/2) = (1/2) * (3) * (x) * ((3sqrt(3))/2)
Simplifying this equation:
(1/4) * (3 - x) * ((3sqrt(3))/2) = (1/2) * (3) * (x) * ((3sqrt(3))/2)
(3 - x) * ((3sqrt(3))/2) = 3 * (x) * ((3sqrt(3))/2)
(3 - x) = 3 * (x)
3 - x = 3x
4x = 3
x = 3/4 cm
Therefore, the length of AD is 3/4 cm.
Area = (1/2) * base * height
Since triangle ABC is an equilateral triangle, all sides have the same length, which is 3 cm. Therefore, the height of triangle ABC is given by:
Height = (sqrt(3)/2) * side = (sqrt(3)/2) * 3 = (3sqrt(3))/2 cm
Let AD = x cm. Then, the base of triangle ADC is BC - AD = 3 - x cm.
The area of triangle ADC is given as half the area of triangle ABD. Therefore:
(1/2) * (1/2) * (3 - x) * ((3sqrt(3))/2) = (1/2) * (3) * (x) * ((3sqrt(3))/2)
Simplifying this equation:
(1/4) * (3 - x) * ((3sqrt(3))/2) = (1/2) * (3) * (x) * ((3sqrt(3))/2)
(3 - x) * ((3sqrt(3))/2) = 3 * (x) * ((3sqrt(3))/2)
(3 - x) = 3 * (x)
3 - x = 3x
4x = 3
x = 3/4 cm
Therefore, the length of AD is 3/4 cm.
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The length of each side of an equilateral triangle ABC is 3 cm. Let D be a point on BC such thatthe area of triangle ADC is half the area of triangle ABD. Then the length of AD. in cm, isa)√7b)√8c)√6d)√5Correct answer is option 'A'. Can you explain this answer?
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