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The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true
a) ar(triangle GBC)=1/2 ar AFGE
b) ar triangle GBC = ar AFGE
c) ar triangle GBC= ar 1/3 AFGE
d) ar triangle GBC= ar 1/4 AFGE?
a) ar(triangle GBC)=1/2 ar AFGE
b) ar triangle GBC = ar AFGE
c) ar triangle GBC= ar 1/3 AFGE
d) ar triangle GBC= ar 1/4 AFGE?
Most Upvoted Answer
The medians BE and CF of a triangle ABC intersect at G. Then which of ...
Understanding the Triangle and Its Medians
In triangle ABC, medians BE and CF intersect at point G, the centroid. The centroid divides each median in a 2:1 ratio.
Area Relationships in Triangle ABC
- The area of triangle ABC can be divided into six smaller triangles by the medians.
- Each median splits the triangle into two triangles of equal area. Thus,
- Area of triangle ABE = Area of triangle AEC
- Area of triangle BCF = Area of triangle BAF, and so forth.
Analyzing Areas Involving G
- The centroid divides each median into segments in the ratio 2:1. Hence:
- Area of triangle GBC = (1/3) * Area of triangle ABC
- Area of triangle AFGE = (1/2) * Area of triangle ABC
Relating the Areas
- Now, we can derive:
- Area of triangle GBC = (1/3) * Area of triangle ABC
- Area of triangle AFGE = (1/2) * Area of triangle ABC
Since the area of triangle GBC is a third of the whole triangle, and the area of quadrilateral AFGE is half of the triangle ABC, we can compare these areas.
Final Comparison
- To find the relationship between the areas, we can express:
- Area of triangle GBC = (1/3) * Area of triangle ABC
- Area of quadrilateral AFGE = (1/2) * Area of triangle ABC
Hence, the area relationship can be stated as:
- Area of triangle GBC = (2/3) * Area of quadrilateral AFGE
Conclusion
From the analysis, the correct option is:
- d) ar triangle GBC = ar 1/3 AFGE
This reflects the established area relationships in triangle ABC with respect to the centroid G.
In triangle ABC, medians BE and CF intersect at point G, the centroid. The centroid divides each median in a 2:1 ratio.
Area Relationships in Triangle ABC
- The area of triangle ABC can be divided into six smaller triangles by the medians.
- Each median splits the triangle into two triangles of equal area. Thus,
- Area of triangle ABE = Area of triangle AEC
- Area of triangle BCF = Area of triangle BAF, and so forth.
Analyzing Areas Involving G
- The centroid divides each median into segments in the ratio 2:1. Hence:
- Area of triangle GBC = (1/3) * Area of triangle ABC
- Area of triangle AFGE = (1/2) * Area of triangle ABC
Relating the Areas
- Now, we can derive:
- Area of triangle GBC = (1/3) * Area of triangle ABC
- Area of triangle AFGE = (1/2) * Area of triangle ABC
Since the area of triangle GBC is a third of the whole triangle, and the area of quadrilateral AFGE is half of the triangle ABC, we can compare these areas.
Final Comparison
- To find the relationship between the areas, we can express:
- Area of triangle GBC = (1/3) * Area of triangle ABC
- Area of quadrilateral AFGE = (1/2) * Area of triangle ABC
Hence, the area relationship can be stated as:
- Area of triangle GBC = (2/3) * Area of quadrilateral AFGE
Conclusion
From the analysis, the correct option is:
- d) ar triangle GBC = ar 1/3 AFGE
This reflects the established area relationships in triangle ABC with respect to the centroid G.
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The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true a) ar(triangle GBC)=1/2 ar AFGEb) ar triangle GBC = ar AFGE c) ar triangle GBC= ar 1/3 AFGEd) ar triangle GBC= ar 1/4 AFGE?
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The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true a) ar(triangle GBC)=1/2 ar AFGEb) ar triangle GBC = ar AFGE c) ar triangle GBC= ar 1/3 AFGEd) ar triangle GBC= ar 1/4 AFGE? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true a) ar(triangle GBC)=1/2 ar AFGEb) ar triangle GBC = ar AFGE c) ar triangle GBC= ar 1/3 AFGEd) ar triangle GBC= ar 1/4 AFGE? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true a) ar(triangle GBC)=1/2 ar AFGEb) ar triangle GBC = ar AFGE c) ar triangle GBC= ar 1/3 AFGEd) ar triangle GBC= ar 1/4 AFGE?.
The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true a) ar(triangle GBC)=1/2 ar AFGEb) ar triangle GBC = ar AFGE c) ar triangle GBC= ar 1/3 AFGEd) ar triangle GBC= ar 1/4 AFGE? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true a) ar(triangle GBC)=1/2 ar AFGEb) ar triangle GBC = ar AFGE c) ar triangle GBC= ar 1/3 AFGEd) ar triangle GBC= ar 1/4 AFGE? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The medians BE and CF of a triangle ABC intersect at G. Then which of the following is true a) ar(triangle GBC)=1/2 ar AFGEb) ar triangle GBC = ar AFGE c) ar triangle GBC= ar 1/3 AFGEd) ar triangle GBC= ar 1/4 AFGE?.
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