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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?
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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.
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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)
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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?
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