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In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.
Q. Which of the following is correct?
- a)(a + b + c) < (p + q + r)
- b)3(a + b + c) < 4(p + q + r)
- c)2(a + b + c) < 3(p + q + r)
- d)3(a + b + c) > 4(p + q + r)
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In a triangle ABC, a, b and c are the lengths of the sides and p, q an...
Let the centroid be O
In triangle AOB
AO + BO > AB
Centroid divides the median in the ratio 2 ∶ 1
∴ 2p/3 + 2q/3 > c
Similarly for other medians
2q/3 + 2r/3 > a
2p/3 + 2r/3 > b
Adding the inequalities
4/3(p + q + r) > (a + b + c)
∴ 4(p + q + r) > 3(a + b + c)
In triangle AOB
AO + BO > AB
Centroid divides the median in the ratio 2 ∶ 1
∴ 2p/3 + 2q/3 > c
Similarly for other medians
2q/3 + 2r/3 > a
2p/3 + 2r/3 > b
Adding the inequalities
4/3(p + q + r) > (a + b + c)
∴ 4(p + q + r) > 3(a + b + c)
Most Upvoted Answer
In a triangle ABC, a, b and c are the lengths of the sides and p, q an...
Explanation:
Properties of Medians in a Triangle:
- In a triangle, a median is a line segment joining a vertex to the midpoint of the opposite side.
- The three medians of a triangle meet at a point called the centroid, which divides each median in a 2:1 ratio.
- The sum of the lengths of any two medians in a triangle is greater than the third median.
Comparison of Sides and Medians:
- Let's consider the triangle ABC with sides a, b, and c, and medians p, q, and r.
- According to the property mentioned earlier, the sum of the lengths of any two medians in a triangle is greater than the third median.
- Therefore, we can write the following inequality: p + q > r, q + r > p, r + p > q.
Answer Explanation:
- To compare the sum of the lengths of sides and medians, we can rewrite the inequalities in terms of sides a, b, and c as follows:
- p + q + r > r, q + r + r > p, r + p + q > q
- Simplifying the above inequalities, we get: p + q + r > r, q + r + r > p, r + p + q > q
- Further simplifying, we get: 2(p + q + r) > a + b + c
- Finally, we can rewrite the above inequality as: a + b + c < 2(p="" +="" q="" +="" />
Therefore, the correct option is option 'B': 3(a + b + c) < 4(p="" +="" q="" +="" />.
Properties of Medians in a Triangle:
- In a triangle, a median is a line segment joining a vertex to the midpoint of the opposite side.
- The three medians of a triangle meet at a point called the centroid, which divides each median in a 2:1 ratio.
- The sum of the lengths of any two medians in a triangle is greater than the third median.
Comparison of Sides and Medians:
- Let's consider the triangle ABC with sides a, b, and c, and medians p, q, and r.
- According to the property mentioned earlier, the sum of the lengths of any two medians in a triangle is greater than the third median.
- Therefore, we can write the following inequality: p + q > r, q + r > p, r + p > q.
Answer Explanation:
- To compare the sum of the lengths of sides and medians, we can rewrite the inequalities in terms of sides a, b, and c as follows:
- p + q + r > r, q + r + r > p, r + p + q > q
- Simplifying the above inequalities, we get: p + q + r > r, q + r + r > p, r + p + q > q
- Further simplifying, we get: 2(p + q + r) > a + b + c
- Finally, we can rewrite the above inequality as: a + b + c < 2(p="" +="" q="" +="" />
Therefore, the correct option is option 'B': 3(a + b + c) < 4(p="" +="" q="" +="" />.
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In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.Q.Which of the following is correct?a)(a + b + c) < (p + q + r)b)3(a + b + c) < 4(p + q + r)c)2(a + b + c) < 3(p + q + r)d)3(a + b + c) > 4(p + q + r)Correct answer is option 'B'. Can you explain this answer?
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In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.Q.Which of the following is correct?a)(a + b + c) < (p + q + r)b)3(a + b + c) < 4(p + q + r)c)2(a + b + c) < 3(p + q + r)d)3(a + b + c) > 4(p + q + r)Correct answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.Q.Which of the following is correct?a)(a + b + c) < (p + q + r)b)3(a + b + c) < 4(p + q + r)c)2(a + b + c) < 3(p + q + r)d)3(a + b + c) > 4(p + q + r)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.Q.Which of the following is correct?a)(a + b + c) < (p + q + r)b)3(a + b + c) < 4(p + q + r)c)2(a + b + c) < 3(p + q + r)d)3(a + b + c) > 4(p + q + r)Correct answer is option 'B'. Can you explain this answer?.
In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.Q.Which of the following is correct?a)(a + b + c) < (p + q + r)b)3(a + b + c) < 4(p + q + r)c)2(a + b + c) < 3(p + q + r)d)3(a + b + c) > 4(p + q + r)Correct answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.Q.Which of the following is correct?a)(a + b + c) < (p + q + r)b)3(a + b + c) < 4(p + q + r)c)2(a + b + c) < 3(p + q + r)d)3(a + b + c) > 4(p + q + r)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC, a, b and c are the lengths of the sides and p, q and r are the lengths of its medians.Q.Which of the following is correct?a)(a + b + c) < (p + q + r)b)3(a + b + c) < 4(p + q + r)c)2(a + b + c) < 3(p + q + r)d)3(a + b + c) > 4(p + q + r)Correct answer is option 'B'. Can you explain this answer?.
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