Question

ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a diagonal then​

Answer 1

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$\large \underline \bold{Correct\:Question}$

ABCD is a quadrilateral in which P, Q, R, and S are mid-points of the lines AB, BC, CD and DA ( see in fig ). AC is a diagonal. Show that:-

• (i) SR || AC and SR =$\dfrac{1}{2}$AC
• (ii) PQ= SR
• (iii) PQRS is a parallelogram.

$\large \underline \bold{Solution:-}$

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

• (i) In △DAC , S is the mid point of DA and R is the mid point of DC. Therefore, SR∥AC and SR=$\dfrac{1}{2}$AC.

By mid-point theorem.

• (ii) In △BAC , P is the mid point of AB and Q is the mid point of BC. Therefore, PQ∥AC and PQ= $\dfrac{1}{2}$AC.

By mid-point theorem.

But from (i) SR =$\dfrac{1}{2}$ AC therefore PQ=SR

• (iii) PQ ∥ AC & SR ∥ AC therefore PQ ∥ SR and PQ = SR.

$\large \underline \bold{Hence,}$

A quadrilateral with opposite sides equal and paralle is a parallelogram. Therefore PQRS is a parallelogram.

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