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In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, is
- a)78
- b)80
- c)72
- d)68
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
In a triangle ABC, medians AD and BE are perpendicular to each other, ...
It is given that AD and BE are medians which are perpendicular to each other.
The lengths of AD and BE are 12cm and 9cm respectively.
It is known that the centroid G divides the median in the ratio of 2:1
Area of ABC = 2* Area of the triangle ABD
Area of ABD = Area of AGB + Area of BGD
The lengths of AD and BE are 12cm and 9cm respectively.
It is known that the centroid G divides the median in the ratio of 2:1
Area of ABC = 2* Area of the triangle ABD
Area of ABD = Area of AGB + Area of BGD
Most Upvoted Answer
In a triangle ABC, medians AD and BE are perpendicular to each other, ...
To find the area of triangle ABC, we need to use the formula for the area of a triangle, which is given by:
Area = (1/2) * base * height
Since the medians AD and BE are perpendicular to each other, we can use the fact that the product of the lengths of two perpendicular segments from a point to the sides of a triangle is equal to the product of the lengths of the other two segments. In this case, we have:
AD * BE = BD * DE + AE * EC
Given that AD = 12 cm and BE = 9 cm, we can substitute these values into the equation:
12 * 9 = BD * DE + AE * EC
Simplifying, we get:
108 = BD * DE + AE * EC
Now, let's calculate the lengths of BD and DE using the fact that BD is half the length of AD, and DE is half the length of BE:
BD = AD/2 = 12/2 = 6 cm
DE = BE/2 = 9/2 = 4.5 cm
Substituting these values into the equation, we have:
108 = 6 * 4.5 + AE * EC
Simplifying further:
108 = 27 + AE * EC
Subtracting 27 from both sides:
AE * EC = 81
Now, let's calculate the area of triangle ABC using the formula mentioned earlier. We can take AE and EC as the base and height of the triangle, respectively:
Area = (1/2) * AE * EC
Substituting the value of AE * EC from the previous equation, we have:
Area = (1/2) * 81 = 40.5 sq cm
So, the area of triangle ABC is 40.5 sq cm. However, this is not one of the given answer options. It appears there may be an error in the question or answer choices provided.
Area = (1/2) * base * height
Since the medians AD and BE are perpendicular to each other, we can use the fact that the product of the lengths of two perpendicular segments from a point to the sides of a triangle is equal to the product of the lengths of the other two segments. In this case, we have:
AD * BE = BD * DE + AE * EC
Given that AD = 12 cm and BE = 9 cm, we can substitute these values into the equation:
12 * 9 = BD * DE + AE * EC
Simplifying, we get:
108 = BD * DE + AE * EC
Now, let's calculate the lengths of BD and DE using the fact that BD is half the length of AD, and DE is half the length of BE:
BD = AD/2 = 12/2 = 6 cm
DE = BE/2 = 9/2 = 4.5 cm
Substituting these values into the equation, we have:
108 = 6 * 4.5 + AE * EC
Simplifying further:
108 = 27 + AE * EC
Subtracting 27 from both sides:
AE * EC = 81
Now, let's calculate the area of triangle ABC using the formula mentioned earlier. We can take AE and EC as the base and height of the triangle, respectively:
Area = (1/2) * AE * EC
Substituting the value of AE * EC from the previous equation, we have:
Area = (1/2) * 81 = 40.5 sq cm
So, the area of triangle ABC is 40.5 sq cm. However, this is not one of the given answer options. It appears there may be an error in the question or answer choices provided.
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In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, isa)78b)80c)72d)68Correct answer is option 'C'. Can you explain this answer?
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