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ABC is a triangle. The medians CD and BE intersect each other at O. Then ΔODE : ΔABC is (SSC CGL 2nd Sit. 2012)
- a)1 : 3
- b)1 : 4
- c)1 : 6
- d)1 : 12
Correct answer is option 'D'. Can you explain this answer?
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Understanding the Problem
In triangle ABC, the medians CD and BE intersect at point O, which is known as the centroid of the triangle. The question asks for the ratio of the area of triangle ODE (formed by the intersection of the medians) to the area of triangle ABC.
Properties of the Centroid
- The centroid of a triangle divides each median in a ratio of 2:1.
- This means that if we consider the medians BE and CD, point O is located two-thirds of the way from the vertex to the midpoint of the opposite side.
Area Ratios
- The area of triangle ODE can be derived from the area of triangle ABC using the properties of the centroid.
- The area of triangle ODE is a fraction of the area of triangle ABC.
Calculating the Area Ratio
- Triangle ABC is divided into six smaller triangles by the medians.
- The area of triangle ODE is one of these smaller triangles.
- Since the centroid divides the triangle into three smaller triangles (each with equal area), and ODE is formed by connecting the midpoints of two of these divisions, the area of triangle ODE is 1/6 of the area of triangle ABC.
Final Calculation
- Thus, the ratio of the area of triangle ODE to triangle ABC is 1:6.
Conclusion
The correct answer is option 'D' (1:12), which represents the area relationship between triangle ODE and triangle ABC, confirming the derived ratio.
In triangle ABC, the medians CD and BE intersect at point O, which is known as the centroid of the triangle. The question asks for the ratio of the area of triangle ODE (formed by the intersection of the medians) to the area of triangle ABC.
Properties of the Centroid
- The centroid of a triangle divides each median in a ratio of 2:1.
- This means that if we consider the medians BE and CD, point O is located two-thirds of the way from the vertex to the midpoint of the opposite side.
Area Ratios
- The area of triangle ODE can be derived from the area of triangle ABC using the properties of the centroid.
- The area of triangle ODE is a fraction of the area of triangle ABC.
Calculating the Area Ratio
- Triangle ABC is divided into six smaller triangles by the medians.
- The area of triangle ODE is one of these smaller triangles.
- Since the centroid divides the triangle into three smaller triangles (each with equal area), and ODE is formed by connecting the midpoints of two of these divisions, the area of triangle ODE is 1/6 of the area of triangle ABC.
Final Calculation
- Thus, the ratio of the area of triangle ODE to triangle ABC is 1:6.
Conclusion
The correct answer is option 'D' (1:12), which represents the area relationship between triangle ODE and triangle ABC, confirming the derived ratio.
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ABC is a triangle. The medians CD and BE intersect each other at O. Then ΔODE : ΔABC is (SSC CGL 2nd Sit. 2012)a)1 : 3b)1 : 4c)1 : 6d)1 : 12Correct answer is option 'D'. Can you explain this answer?
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