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Question
ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that:
- SR || AC and SR = `1/2AC`
- PQ = SR
- PQRS is a parallelogram.
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Solution
(i) In ΔADC, S and R are the mid-points of sides AD and CD respectively.
In a triangle, the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.
∴ SR || AC and SR = `1/2AC` ...(1)
(ii) In ΔABC, P and Q are mid-points of sides AB and BC respectively. Therefore, by using the mid-point theorem,
PQ || AC and PQ = `1/2AC` ...(2)
Using equations (1) and (2), we obtain
PQ || SR and PQ = SR ...(3)
⇒ PQ = SR
(iii) From equation (3), we obtained
PQ || SR and PQ = SR
Clearly, one pair of opposite sides of quadrilateral PQRS is parallel and equal.
Hence, PQRS is a parallelogram.
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