Class 10 Exam > Class 10 Questions > In the parallelogram ABCD, M and N are respec...
Start Learning for Free
In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratio
- a)1 : 4
- b)1 : 6
- c)1 : 8
- d)1 : 9
Correct answer is option 'C'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
In the parallelogram ABCD, M and N are respectively the midpoints of A...
Given, parallelogram ABCD and M, N are the midpoints of AB and AD respectively. Let's find the area of triangle AMN and parallelogram ABCD.
Finding the area of triangle AMN:
- Since M and N are midpoints, we know that AM = MB and AN = ND.
- Therefore, MN is parallel to BD and MN = 1/2 * BD.
- Also, since AM = MB, angle MAN = angle MBN (alternate interior angles).
- Similarly, since AN = ND, angle MAN = angle NDA (alternate interior angles).
- Therefore, angle MBN = angle NDA, and we can conclude that triangles MBN and NDA are similar.
- Using the similarity, we can write: BN/AD = MB/ND = 1/2.
- Therefore, BN = 1/2 * AD and MB = 1/2 * AB.
- Now, the area of triangle AMN can be found using the formula: 1/2 * base * height.
- The base of triangle AMN is MN = 1/2 * BD, and the height is MB = 1/2 * AB.
- Therefore, the area of triangle AMN = 1/2 * (1/2 * BD) * (1/2 * AB) = 1/8 * AB * BD.
Finding the area of parallelogram ABCD:
- The area of a parallelogram can be found using the formula: base * height.
- The base of parallelogram ABCD is AB, and the height is the perpendicular distance between AB and CD.
- Since AB is parallel to CD, we can take the perpendicular distance as the length of any of the altitudes.
- Let's take the altitude from A to CD. Since AD is parallel to BC, we know that the altitude from A to CD is equal to the altitude from D to AB.
- Let's call the altitude h. Then, the area of parallelogram ABCD = AB * h.
Comparing the ratios:
- We need to find the ratio of the area of triangle AMN to the area of parallelogram ABCD.
- From the above calculations, we have: area of triangle AMN = 1/8 * AB * BD and area of parallelogram ABCD = AB * h.
- Therefore, the ratio of the areas = (1/8 * AB * BD)/(AB * h) = 1/8h.
- We know that h = AD * sin(angle A) = AD * sin(angle D) (since opposite angles are equal in a parallelogram).
- Also, in triangle ABD, sin(angle D) = BD/AB (opposite/hypotenuse).
- Therefore, h = AD * BD/AB.
- Substituting this in the ratio of areas, we get: ratio = 1/8 * AB/(AD * BD/AB) = 1/8 * AB^2/(AB * AD * BD) = 1/8 * 1/AD.
- Therefore, the ratio of the areas of triangle AMN and parallelogram ABCD = 1 : 8.
- Hence, the correct option is C, i.e., 1 : 8.
Finding the area of triangle AMN:
- Since M and N are midpoints, we know that AM = MB and AN = ND.
- Therefore, MN is parallel to BD and MN = 1/2 * BD.
- Also, since AM = MB, angle MAN = angle MBN (alternate interior angles).
- Similarly, since AN = ND, angle MAN = angle NDA (alternate interior angles).
- Therefore, angle MBN = angle NDA, and we can conclude that triangles MBN and NDA are similar.
- Using the similarity, we can write: BN/AD = MB/ND = 1/2.
- Therefore, BN = 1/2 * AD and MB = 1/2 * AB.
- Now, the area of triangle AMN can be found using the formula: 1/2 * base * height.
- The base of triangle AMN is MN = 1/2 * BD, and the height is MB = 1/2 * AB.
- Therefore, the area of triangle AMN = 1/2 * (1/2 * BD) * (1/2 * AB) = 1/8 * AB * BD.
Finding the area of parallelogram ABCD:
- The area of a parallelogram can be found using the formula: base * height.
- The base of parallelogram ABCD is AB, and the height is the perpendicular distance between AB and CD.
- Since AB is parallel to CD, we can take the perpendicular distance as the length of any of the altitudes.
- Let's take the altitude from A to CD. Since AD is parallel to BC, we know that the altitude from A to CD is equal to the altitude from D to AB.
- Let's call the altitude h. Then, the area of parallelogram ABCD = AB * h.
Comparing the ratios:
- We need to find the ratio of the area of triangle AMN to the area of parallelogram ABCD.
- From the above calculations, we have: area of triangle AMN = 1/8 * AB * BD and area of parallelogram ABCD = AB * h.
- Therefore, the ratio of the areas = (1/8 * AB * BD)/(AB * h) = 1/8h.
- We know that h = AD * sin(angle A) = AD * sin(angle D) (since opposite angles are equal in a parallelogram).
- Also, in triangle ABD, sin(angle D) = BD/AB (opposite/hypotenuse).
- Therefore, h = AD * BD/AB.
- Substituting this in the ratio of areas, we get: ratio = 1/8 * AB/(AD * BD/AB) = 1/8 * AB^2/(AB * AD * BD) = 1/8 * 1/AD.
- Therefore, the ratio of the areas of triangle AMN and parallelogram ABCD = 1 : 8.
- Hence, the correct option is C, i.e., 1 : 8.
Free Test
FREE
| Start Free Test |
Community Answer
In the parallelogram ABCD, M and N are respectively the midpoints of A...
P is mid point of CD.
ar(ANM) = (1/4)(ar)(AMPD)
=1/8 ar(ABCD)
ar(ANM) = (1/4)(ar)(AMPD)
=1/8 ar(ABCD)
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
|
Explore Courses for Class 10 exam
|
|
Similar Class 10 Doubts
Top Courses for Class 10View all
In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer?
Question Description
In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer?.
In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer?.
Solutions for In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer?, a detailed solution for In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice In the parallelogram ABCD, M and N are respectively the midpoints of AB and AD. The points M and N are joined to form the triangle AMN. The area of the triangle AMN and the area of the parallelogram ABCD are in the ratioa)1 : 4b)1 : 6c)1 : 8d)1 : 9Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Suggested Free Tests
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!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup