Question

${\displaystyle \square }$ABCD is a parallelogram, P and Q are midpoints of side AB and DC respectively, then prove ${\displaystyle \square }$APCQ is a parallelogram.

Answer 1

Given: ${\displaystyle \square }$ABCD is a parallelogram. P and Q are the midpoints of sides AB and DC respectively.

To prove: ${\displaystyle \square }$APCQ is a parallelogram.

Proof:

AP = ${\displaystyle \frac{1}{2}}$ AB    …(i) [P is the midpoint of side AB]

QC = ${\displaystyle \frac{1}{2}}$ DC     …(ii) [Q is the midpoint of side CD]

${\displaystyle \square }$ABCD is a parallelogram.     ...[Given]

∴ AB = DC       ...[Opposite sides of a parallelogram]

∴ ${\displaystyle \frac{1}{2}}$ AB = ${\displaystyle \frac{1}{2}}$ DC     ...[Multiplying both sides by ${\displaystyle \frac{1}{2}}$]

∴ AP = QC      …(iii) [From (i) and (ii)]

Also, AB || DC    ...[Opposite angles of a parallelogram]

i.e. AP || QC     …(iv) [A–P–B, D–Q–C]

From (iii) and (iv),

AP = QC

AP || QC

A quadrilateral is a parallelogram if its opposite sides is parallel and congruent.

∴ ${\displaystyle \square }$APCQ is a parallelogram.