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ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD is
- a)1:4
- b)2:3
- c)1:2
- d)1:3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
ABCD is a parallelogram in which diagonals AC and BD intersect at O. I...
In Δ OAB,
Mid- point of OA = E
Mid- point of OB = H
Therefore,
EH|| AB
Therefore,
HE = AB/2
Similarly,
HG = BC/2
FG = CD/2
EF = AD/2
EF + HG + FG + EF = (1/2)(AB + BC + CD + AD)
⇒ Perimeter of EFGH = (1/2) perimeter of ABCD
So,
The ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD = 1:2
Most Upvoted Answer
ABCD is a parallelogram in which diagonals AC and BD intersect at O. I...
Given:
- ABCD is a parallelogram.
- Diagonals AC and BD intersect at point O.
- E, F, G, and H are the midpoints of AO, DO, CO, and BO respectively.
To find:
The ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD.
Solution:
Step 1: Understanding the problem:
- Let's first understand the properties of a parallelogram and its diagonals.
- In a parallelogram, opposite sides are equal and parallel.
- The diagonals of a parallelogram bisect each other.
- The diagonals of a parallelogram divide it into four triangles of equal area.
- The diagonals of a parallelogram also divide it into four smaller parallelograms of equal area.
Step 2: Drawing the diagram:
- Draw a parallelogram ABCD.
- Draw the diagonals AC and BD intersecting at point O.
- Label the midpoints of AO, DO, CO, and BO as E, F, G, and H respectively.
Step 3: Finding the ratio of perimeters:
- Let's find the lengths of the sides of the quadrilateral EFGH and the parallelogram ABCD.
- In parallelogram ABCD, opposite sides are equal. Let's assume the length of AB or DC as 'a' and the length of BC or AD as 'b'.
- Since E, F, G, and H are midpoints, the length of EO or OA is half the length of AO, which is 'a/2'.
- Similarly, the length of FO or OD is 'b/2', the length of GO or OC is 'a/2', and the length of HO or OB is 'b/2'.
- The perimeter of quadrilateral EFGH is EF + FG + GH + HE.
- Substituting the lengths we found, the perimeter of EFGH is (a/2 + b/2) + (b/2 + a/2) + (a/2 + b/2) + (b/2 + a/2) = 2a + 2b.
- The perimeter of parallelogram ABCD is AB + BC + CD + DA. Substituting the lengths, the perimeter of ABCD is a + b + a + b = 2a + 2b.
- Therefore, the ratio of the perimeter of EFGH to the perimeter of ABCD is (2a + 2b) / (2a + 2b) = 1:1.
- Simplifying, we get the ratio as 1:1, which is the same as 1:2 when simplified further.
- Hence, the correct answer is option 'C'.
Step 4: Answer:
The ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD is 1:2.
- ABCD is a parallelogram.
- Diagonals AC and BD intersect at point O.
- E, F, G, and H are the midpoints of AO, DO, CO, and BO respectively.
To find:
The ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD.
Solution:
Step 1: Understanding the problem:
- Let's first understand the properties of a parallelogram and its diagonals.
- In a parallelogram, opposite sides are equal and parallel.
- The diagonals of a parallelogram bisect each other.
- The diagonals of a parallelogram divide it into four triangles of equal area.
- The diagonals of a parallelogram also divide it into four smaller parallelograms of equal area.
Step 2: Drawing the diagram:
- Draw a parallelogram ABCD.
- Draw the diagonals AC and BD intersecting at point O.
- Label the midpoints of AO, DO, CO, and BO as E, F, G, and H respectively.
Step 3: Finding the ratio of perimeters:
- Let's find the lengths of the sides of the quadrilateral EFGH and the parallelogram ABCD.
- In parallelogram ABCD, opposite sides are equal. Let's assume the length of AB or DC as 'a' and the length of BC or AD as 'b'.
- Since E, F, G, and H are midpoints, the length of EO or OA is half the length of AO, which is 'a/2'.
- Similarly, the length of FO or OD is 'b/2', the length of GO or OC is 'a/2', and the length of HO or OB is 'b/2'.
- The perimeter of quadrilateral EFGH is EF + FG + GH + HE.
- Substituting the lengths we found, the perimeter of EFGH is (a/2 + b/2) + (b/2 + a/2) + (a/2 + b/2) + (b/2 + a/2) = 2a + 2b.
- The perimeter of parallelogram ABCD is AB + BC + CD + DA. Substituting the lengths, the perimeter of ABCD is a + b + a + b = 2a + 2b.
- Therefore, the ratio of the perimeter of EFGH to the perimeter of ABCD is (2a + 2b) / (2a + 2b) = 1:1.
- Simplifying, we get the ratio as 1:1, which is the same as 1:2 when simplified further.
- Hence, the correct answer is option 'C'.
Step 4: Answer:
The ratio of the perimeter of quadrilateral EFGH to the perimeter of parallelogram ABCD is 1:2.
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ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD isa)1:4b)2:3c)1:2d)1:3Correct answer is option 'C'. Can you explain this answer?
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ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD isa)1:4b)2:3c)1:2d)1:3Correct answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD isa)1:4b)2:3c)1:2d)1:3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD isa)1:4b)2:3c)1:2d)1:3Correct answer is option 'C'. Can you explain this answer?.
ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD isa)1:4b)2:3c)1:2d)1:3Correct answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD isa)1:4b)2:3c)1:2d)1:3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a parallelogram in which diagonals AC and BD intersect at O. If E, F, G and H are the mid points of AO, DO, CO and BO respectively, then the ratio of the perimeter of the quadrilateral EFGH to the perimeter of parallelogram ABCD isa)1:4b)2:3c)1:2d)1:3Correct answer is option 'C'. Can you explain this answer?.
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