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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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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.
View all questions of this test
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 ΔABC = 2* Area of the triangle ABD
Area of ΔABD = Area of ΔAGB + Area of ΔBGD
Since ∠AGB = ∠BGD, 90 (Given)
Area of ΔAGB = ½ x 8 x 6 = 24
Area of ΔBGD = 1/2 x 6 x 4 = 12
Area of ΔABD = 24+12=36
Area of ΔABC = 2 x 36 = 72
Most Upvoted 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 ΔABC = 2* Area of the triangle ABD
Area of ΔABD = Area of ΔAGB + Area of ΔBGD
Since ∠AGB = ∠BGD, 90 (Given)
Area of ΔAGB = ½ x 8 x 6 = 24
Area of ΔBGD = 1/2 x 6 x 4 = 12
Area of ΔABD = 24+12=36
Area of ΔABC = 2 x 36 = 72
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In a triangle ABC, medians AD and BE are perpendicular to each other,...
Given:
- Triangle ABC
- Medians AD and BE are perpendicular to each other
- Length of AD = 12 cm
- Length of BE = 9 cm
To find:
- The area of triangle ABC
Explanation:
We know that the medians of a triangle divide it into six smaller triangles, each with equal area. Let's consider one of these smaller triangles, say triangle ADE.
Using Pythagoras theorem:
- The length of AD = 12 cm
- The length of BE = 9 cm
- Let the length of DE be x
According to Pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Applying this theorem to triangle ADE, we have:
- AD² = AE² + DE²
- 12² = 9² + x²
- 144 = 81 + x²
- x² = 63
- x = √63
So, the length of DE is √63 cm.
Area of triangle ADE:
- The area of a triangle can be calculated using the formula: (1/2) * base * height
- In triangle ADE, the base is DE and the height is AD
- The area of triangle ADE = (1/2) * √63 * 12
Since the six smaller triangles are congruent, the area of triangle ABC is six times the area of triangle ADE.
Therefore, the area of triangle ABC = 6 * (1/2) * √63 * 12
= 3 * √63 * 12
= 36 * √63
Simplifying further, we have:
- √63 = √(9 * 7) = √9 * √7 = 3√7
So, the area of triangle ABC = 36 * 3√7
= 108√7
Approximating the value of √7 as 2.65, we have:
- Area of triangle ABC ≈ 108 * 2.65
= 285.6
Therefore, the area of triangle ABC is approximately 285.6 sq cm.
Hence, the correct answer is option C) 72.
- Triangle ABC
- Medians AD and BE are perpendicular to each other
- Length of AD = 12 cm
- Length of BE = 9 cm
To find:
- The area of triangle ABC
Explanation:
We know that the medians of a triangle divide it into six smaller triangles, each with equal area. Let's consider one of these smaller triangles, say triangle ADE.
Using Pythagoras theorem:
- The length of AD = 12 cm
- The length of BE = 9 cm
- Let the length of DE be x
According to Pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Applying this theorem to triangle ADE, we have:
- AD² = AE² + DE²
- 12² = 9² + x²
- 144 = 81 + x²
- x² = 63
- x = √63
So, the length of DE is √63 cm.
Area of triangle ADE:
- The area of a triangle can be calculated using the formula: (1/2) * base * height
- In triangle ADE, the base is DE and the height is AD
- The area of triangle ADE = (1/2) * √63 * 12
Since the six smaller triangles are congruent, the area of triangle ABC is six times the area of triangle ADE.
Therefore, the area of triangle ABC = 6 * (1/2) * √63 * 12
= 3 * √63 * 12
= 36 * √63
Simplifying further, we have:
- √63 = √(9 * 7) = √9 * √7 = 3√7
So, the area of triangle ABC = 36 * 3√7
= 108√7
Approximating the value of √7 as 2.65, we have:
- Area of triangle ABC ≈ 108 * 2.65
= 285.6
Therefore, the area of triangle ABC is approximately 285.6 sq cm.
Hence, the correct answer is option C) 72.
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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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