Defence Exam > Defence Questions > Consider the following statements:1) The peri...
Start Learning for Free
Consider the following statements:
1) The perimeter of a triangle is greater than the sum of its three medinas.
2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.
Which of the above statements is/are correct?
1) The perimeter of a triangle is greater than the sum of its three medinas.
2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.
Which of the above statements is/are correct?
- a)1 only
- b)2 only
- c)Both 1 and 2
- d)Neither 1 nor 2
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Consider the following statements:1) The perimeter of a triangle is gr...
1. Perimeter of triangle is greater than the sum of 3 medians
Let ABC be the triangle and D. E and F are midpoints of BC, CA and AB respectively.
Recall that the sum of two sides of a triangle is greater than twice the median bisecting the third side,(Theorem to be remembered)
Hence in ΔABD, AD is a median
⇒ AB + AC > 2(AD)
Similarly, we get
BC + AC > 2CF
BC + AB > 2BE
On adding the above inequations, we get
(AB + AC) + (BC + AC) + (BC + AB )> 2AD + 2CD + 2BE
2(AB + BC + AC) > 2(AD + BE + CF)
∴ AB + BC + AC > AD + BE + CF
2.
To prove: AB + BC + CA > 2AD
Construction: AD is joined
Proof: In triangle ABD, AB + BD > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 1
In triangle ADC, AC + DC > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 2
Adding 1 and 2 we get,
AB + BD + AC + DC > AD + AD
=> AB + (BD + DC) + AC > 2AD
=> AB + BC + AC > 2AD
Hence proved
Let ABC be the triangle and D. E and F are midpoints of BC, CA and AB respectively.
Recall that the sum of two sides of a triangle is greater than twice the median bisecting the third side,(Theorem to be remembered)
Hence in ΔABD, AD is a median
⇒ AB + AC > 2(AD)
Similarly, we get
BC + AC > 2CF
BC + AB > 2BE
On adding the above inequations, we get
(AB + AC) + (BC + AC) + (BC + AB )> 2AD + 2CD + 2BE
2(AB + BC + AC) > 2(AD + BE + CF)
∴ AB + BC + AC > AD + BE + CF
2.
To prove: AB + BC + CA > 2AD
Construction: AD is joined
Proof: In triangle ABD, AB + BD > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 1
In triangle ADC, AC + DC > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 2
Adding 1 and 2 we get,
AB + BD + AC + DC > AD + AD
=> AB + (BD + DC) + AC > 2AD
=> AB + BC + AC > 2AD
Hence proved
Most Upvoted Answer
Consider the following statements:1) The perimeter of a triangle is gr...
Explanation of Statement 1
The first statement asserts that the perimeter of a triangle is greater than the sum of its three medians.
- Understanding Medians: A median of a triangle connects a vertex to the midpoint of the opposite side. Each triangle has three medians.
- Perimeter vs. Medians: The perimeter of a triangle (sum of all sides) is indeed greater than the sum of its medians. This can be shown mathematically or geometrically, as the medians are always shorter than the sides they connect.
- Conclusion: Hence, Statement 1 is correct.
Explanation of Statement 2
The second statement claims that for any triangle ABC with a point D on side BC, the inequality AB + BC + CA > 2AD holds.
- Understanding the Inequality: This statement reflects a property of triangles related to distances. The sum of the lengths of two sides of a triangle (here, AB and AC) compared to twice the length from a vertex to a point on the opposite side (AD) indicates that traveling along the triangle's perimeter is always shorter than directly connecting across.
- Proof by Triangle Inequality: The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This property supports Statement 2 as true.
- Conclusion: Therefore, Statement 2 is also correct.
Final Conclusion
Since both statements are correct, the correct answer is option 'C': Both 1 and 2.
The first statement asserts that the perimeter of a triangle is greater than the sum of its three medians.
- Understanding Medians: A median of a triangle connects a vertex to the midpoint of the opposite side. Each triangle has three medians.
- Perimeter vs. Medians: The perimeter of a triangle (sum of all sides) is indeed greater than the sum of its medians. This can be shown mathematically or geometrically, as the medians are always shorter than the sides they connect.
- Conclusion: Hence, Statement 1 is correct.
Explanation of Statement 2
The second statement claims that for any triangle ABC with a point D on side BC, the inequality AB + BC + CA > 2AD holds.
- Understanding the Inequality: This statement reflects a property of triangles related to distances. The sum of the lengths of two sides of a triangle (here, AB and AC) compared to twice the length from a vertex to a point on the opposite side (AD) indicates that traveling along the triangle's perimeter is always shorter than directly connecting across.
- Proof by Triangle Inequality: The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This property supports Statement 2 as true.
- Conclusion: Therefore, Statement 2 is also correct.
Final Conclusion
Since both statements are correct, the correct answer is option 'C': Both 1 and 2.
Free Test
FREE
| Start Free Test |
Community Answer
Consider the following statements:1) The perimeter of a triangle is gr...
1. Perimeter of triangle is greater than the sum of 3 medians
Let ABC be the triangle and D. E and F are midpoints of BC, CA and AB respectively.
Recall that the sum of two sides of a triangle is greater than twice the median bisecting the third side,(Theorem to be remembered)
Hence in ΔABD, AD is a median
⇒ AB + AC > 2(AD)
Similarly, we get
BC + AC > 2CF
BC + AB > 2BE
On adding the above inequations, we get
(AB + AC) + (BC + AC) + (BC + AB )> 2AD + 2CD + 2BE
2(AB + BC + AC) > 2(AD + BE + CF)
∴ AB + BC + AC > AD + BE + CF
2.
To prove: AB + BC + CA > 2AD
Construction: AD is joined
Proof: In triangle ABD, AB + BD > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 1
In triangle ADC, AC + DC > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 2
Adding 1 and 2 we get,
AB + BD + AC + DC > AD + AD
=> AB + (BD + DC) + AC > 2AD
=> AB + BC + AC > 2AD
Hence proved
Let ABC be the triangle and D. E and F are midpoints of BC, CA and AB respectively.
Recall that the sum of two sides of a triangle is greater than twice the median bisecting the third side,(Theorem to be remembered)
Hence in ΔABD, AD is a median
⇒ AB + AC > 2(AD)
Similarly, we get
BC + AC > 2CF
BC + AB > 2BE
On adding the above inequations, we get
(AB + AC) + (BC + AC) + (BC + AB )> 2AD + 2CD + 2BE
2(AB + BC + AC) > 2(AD + BE + CF)
∴ AB + BC + AC > AD + BE + CF
2.
To prove: AB + BC + CA > 2AD
Construction: AD is joined
Proof: In triangle ABD, AB + BD > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 1
In triangle ADC, AC + DC > AD [because, the sum of any two sides of a triangle is always greater than the third side] ---- 2
Adding 1 and 2 we get,
AB + BD + AC + DC > AD + AD
=> AB + (BD + DC) + AC > 2AD
=> AB + BC + AC > 2AD
Hence proved
|
Explore Courses for Defence exam
|
|
Top Courses for DefenceView all
Question Description
Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? for Defence 2025 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer?.
Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? for Defence 2025 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer?.
Solutions for Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Defence.
Download more important topics, notes, lectures and mock test series for Defence Exam by signing up for free.
Here you can find the meaning of Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer?, a detailed solution for Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider the following statements:1) The perimeter of a triangle is greater than the sum of its three medinas.2) In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD.Which of the above statements is/are correct?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice Defence tests.
|
|
Explore Courses for Defence exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup