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The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.?
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The side AB, BC and CA of triangle are extended to D,E and F respectiv...
Given:
- Triangle ABC with sides AB, BC, and CA
- The sides AB, BC, and CA are extended to points D, E, and F respectively such that AB = BD, BC = CE, and CA = AF
To Find:
The ratio of the area of triangle ABC to the area of triangle DEF
Solution:
1. Identify Key Information:
- The sides AB, BC, and CA are extended to points D, E, and F respectively, creating triangle DEF.
- The sides AD, BE, and CF are parallel to the sides BC, CA, and AB respectively.
- The triangle ABC and triangle DEF share the same base, which is the line segment EF.
2. Identify Similar Triangles:
- Since AD || BC and BD || AC, triangle ADB and triangle ABC are similar.
- Similarly, triangle BEC and triangle ABC are similar, and triangle CFA and triangle ABC are similar.
3. Identify Ratios:
- From the similarity of triangle ADB and triangle ABC, we can identify that the ratio of the lengths of the corresponding sides is AD/AB = BD/BC = 1/2.
- Similarly, the ratio of the lengths of the corresponding sides in triangle BEC and triangle ABC is BE/BC = CE/CA = 1/2, and in triangle CFA and triangle ABC is CF/CA = AF/AB = 1/2.
4. Identify Ratios of Areas:
- The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides.
- Therefore, the ratio of the area of triangle ADB to the area of triangle ABC is (AD/AB)^2 = (1/2)^2 = 1/4.
- Similarly, the ratio of the area of triangle BEC to the area of triangle ABC is (BE/BC)^2 = (1/2)^2 = 1/4, and the ratio of the area of triangle CFA to the area of triangle ABC is (CF/CA)^2 = (1/2)^2 = 1/4.
5. Identify the Ratio of the Areas:
- The area of triangle ABC is the sum of the areas of triangle ADB, triangle BEC, and triangle CFA.
- Therefore, the ratio of the area of triangle ABC to the area of triangle DEF is:
(Area of triangle ABC) / (Area of triangle DEF) = [(Area of triangle ADB) + (Area of triangle BEC) + (Area of triangle CFA)] / (Area of triangle DEF)
= [(1/4) + (1/4) + (1/4)] / 1
= 3/4
Answer:
The ratio of the area of triangle ABC to the area of triangle DEF is 3:4.
- Triangle ABC with sides AB, BC, and CA
- The sides AB, BC, and CA are extended to points D, E, and F respectively such that AB = BD, BC = CE, and CA = AF
To Find:
The ratio of the area of triangle ABC to the area of triangle DEF
Solution:
1. Identify Key Information:
- The sides AB, BC, and CA are extended to points D, E, and F respectively, creating triangle DEF.
- The sides AD, BE, and CF are parallel to the sides BC, CA, and AB respectively.
- The triangle ABC and triangle DEF share the same base, which is the line segment EF.
2. Identify Similar Triangles:
- Since AD || BC and BD || AC, triangle ADB and triangle ABC are similar.
- Similarly, triangle BEC and triangle ABC are similar, and triangle CFA and triangle ABC are similar.
3. Identify Ratios:
- From the similarity of triangle ADB and triangle ABC, we can identify that the ratio of the lengths of the corresponding sides is AD/AB = BD/BC = 1/2.
- Similarly, the ratio of the lengths of the corresponding sides in triangle BEC and triangle ABC is BE/BC = CE/CA = 1/2, and in triangle CFA and triangle ABC is CF/CA = AF/AB = 1/2.
4. Identify Ratios of Areas:
- The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides.
- Therefore, the ratio of the area of triangle ADB to the area of triangle ABC is (AD/AB)^2 = (1/2)^2 = 1/4.
- Similarly, the ratio of the area of triangle BEC to the area of triangle ABC is (BE/BC)^2 = (1/2)^2 = 1/4, and the ratio of the area of triangle CFA to the area of triangle ABC is (CF/CA)^2 = (1/2)^2 = 1/4.
5. Identify the Ratio of the Areas:
- The area of triangle ABC is the sum of the areas of triangle ADB, triangle BEC, and triangle CFA.
- Therefore, the ratio of the area of triangle ABC to the area of triangle DEF is:
(Area of triangle ABC) / (Area of triangle DEF) = [(Area of triangle ADB) + (Area of triangle BEC) + (Area of triangle CFA)] / (Area of triangle DEF)
= [(1/4) + (1/4) + (1/4)] / 1
= 3/4
Answer:
The ratio of the area of triangle ABC to the area of triangle DEF is 3:4.
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The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.?
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The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.?.
The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The side AB, BC and CA of triangle are extended to D,E and F respectively such that AB=BD , BC=CE and CA=AF . Find the ratio of area of triangle ABC to that of trangle DEF.?.
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