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The quadrilateral formed by joining the mid-points of a given quadrilateral will be (surely) :
- a)Parallelogram
- b)Rectangle
- c)Rhombus
- d)Square
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The quadrilateral formed by joining the mid-points of a given quadrila...
ABCD is a parallelogram and P,Q,R,S are the midpoints of AB, BC, CD, DA respectively. Consider, AC as a diagonal of ABCD.
Now, According to midpoint theorem,
PQ =1/2 AC and PQ is parallel to AC,
and SR = 1/2 AC and PQ is parallel to AC.
∴ PQ = RS and PQ || RS || AC.
∴ PQRS will be a parallelogram.
Most Upvoted Answer
The quadrilateral formed by joining the mid-points of a given quadrila...
Properties of the Quadrilateral formed by joining the midpoints of a given quadrilateral:
When we join the midpoints of the sides of a quadrilateral, a new quadrilateral is formed. Let's call the given quadrilateral ABCD, and the midpoints of the sides as E, F, G, and H, respectively. The new quadrilateral formed is labeled EFGH.
Proof:
1. Opposite Sides are Parallel:
By joining the midpoints of the sides, we can observe that opposite sides of the new quadrilateral EFGH are parallel to each other. To prove this, we can use the concept of midpoints.
- Segment EF is parallel to segment AB because E and F are the midpoints of AB and CD, respectively.
- Similarly, segment GH is parallel to segment AB.
- Segment FG is parallel to segment BC.
- Lastly, segment EH is parallel to segment BC.
Thus, the opposite sides of the quadrilateral EFGH are parallel to each other.
2. Equal Lengths of Opposite Sides:
We can also observe that the opposite sides of the quadrilateral EFGH are equal in length. This can be proven using the concept of midpoints.
- Segments EF and GH are equal in length because E and F are the midpoints of AB and CD, while G and H are the midpoints of BC and AD, respectively.
- Similarly, segments FG and EH are equal in length.
Hence, the quadrilateral EFGH has opposite sides of equal length.
Therefore, based on the above properties, we can conclude that the quadrilateral formed by joining the midpoints of a given quadrilateral is a parallelogram (option A).
When we join the midpoints of the sides of a quadrilateral, a new quadrilateral is formed. Let's call the given quadrilateral ABCD, and the midpoints of the sides as E, F, G, and H, respectively. The new quadrilateral formed is labeled EFGH.
Proof:
1. Opposite Sides are Parallel:
By joining the midpoints of the sides, we can observe that opposite sides of the new quadrilateral EFGH are parallel to each other. To prove this, we can use the concept of midpoints.
- Segment EF is parallel to segment AB because E and F are the midpoints of AB and CD, respectively.
- Similarly, segment GH is parallel to segment AB.
- Segment FG is parallel to segment BC.
- Lastly, segment EH is parallel to segment BC.
Thus, the opposite sides of the quadrilateral EFGH are parallel to each other.
2. Equal Lengths of Opposite Sides:
We can also observe that the opposite sides of the quadrilateral EFGH are equal in length. This can be proven using the concept of midpoints.
- Segments EF and GH are equal in length because E and F are the midpoints of AB and CD, while G and H are the midpoints of BC and AD, respectively.
- Similarly, segments FG and EH are equal in length.
Hence, the quadrilateral EFGH has opposite sides of equal length.
Therefore, based on the above properties, we can conclude that the quadrilateral formed by joining the midpoints of a given quadrilateral is a parallelogram (option A).
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