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प्रश्न
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
उत्तर
Let ABCD is a quadrilateral in which P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA respectively. Join PQ, QR, RS, SP, and BD.
In ΔABD, S and P are the mid-points of AD and AB respectively. Therefore, by using mid-point theorem, it can be said that
SP || BD and SP = 1/2BD ... (1)
Similarly in ΔBCD,
QR || BD and QR = 1/2BD ... (2)
From equations (1) and (2), we obtain
SP || QR and SP = QR
In quadrilateral SPQR, one pair of opposite sides is equal and parallel to
each other. Therefore, SPQR is a parallelogram.
We know that diagonals of a parallelogram bisect each other.
Hence, PR and QS bisect each other.
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संबंधित प्रश्न
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.
Let Abc Be an Isosceles Triangle in Which Ab = Ac. If D, E, F Be the Mid-points of the Sides Bc, Ca and a B Respectively, Show that the Segment Ad and Ef Bisect Each Other at Right Angles.
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Prove that : (i) AP = 2AR
(ii) BC = 4QR
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