In parallelogram ABCD, X and Y are mid-points of opposite sides AB and DC respectively. Prove that :
(i) AX = YC
(ii) AX is parallel to YC
(iii) AXCY is a parallelogram

Question

In parallelogram ABCD, X and Y are mid-points of opposite sides AB and DC respectively. Prove that : (i) AX = YC (ii) AX is parallel to YC (iii) AXCY is a parallelogram

KnowledgeBoat · Accepted Answer

(i) We know that, Opposite sides of || gm are equal. ∴ AB = CD ⇒ ⇒ AX = CY (As, X and Y are mid-points of AB and CD respectively) Hence, proved that AX = YC. (ii) We know that, Opposite sides of || gm are parallel. ∴ AB || DC ∴ AX || YC. Hence, proved that AX || YC. (iii) From figure, AX = YC and AX || YC. Since, one pair of opposite sides of quadrilateral AXCY are equal and parallel. ∴ AXCY is a || gm. Hence, proved that AXCY is a parallelogram.

Mathematics

In parallelogram ABCD, X and Y are mid-points of opposite sides AB and DC respectively. Prove that :
(i) AX = YC
(ii) AX is parallel to YC
(iii) AXCY is a parallelogram

Rectilinear Figures

Related Questions

The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.

In the following figure, ABCD and PQRS are two parallelograms such that ∠D = 120° and ∠Q = 70°. Find the value of x.

In the following figure, ABCD is a rhombus and DCFE is a square.
If ∠ABC = 56°, find :
(i) ∠DAE
(ii) ∠FEA
(iii) ∠EAC
(iv) ∠AEC

In parallelogram ABCD, E is the mid-point of side AB and CE bisects angle BCD. Prove that :
(i) AE = AD
(ii) DE bisects angle ADC
(iii) angle DEC is a right angle

Mathematics

In parallelogram ABCD, X and Y are mid-points of opposite sides AB and DC respectively. Prove that :(i) AX = YC(ii) AX is parallel to YC(iii) AXCY is a parallelogram

Rectilinear Figures

Related Questions

The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.

In the following figure, ABCD and PQRS are two parallelograms such that ∠D = 120° and ∠Q = 70°. Find the value of x.

In the following figure, ABCD is a rhombus and DCFE is a square.If ∠ABC = 56°, find :(i) ∠DAE(ii) ∠FEA(iii) ∠EAC(iv) ∠AEC

In parallelogram ABCD, E is the mid-point of side AB and CE bisects angle BCD. Prove that :(i) AE = AD(ii) DE bisects angle ADC(iii) angle DEC is a right angle

In parallelogram ABCD, X and Y are mid-points of opposite sides AB and DC respectively. Prove that :
(i) AX = YC
(ii) AX is parallel to YC
(iii) AXCY is a parallelogram

In the following figure, ABCD is a rhombus and DCFE is a square.
If ∠ABC = 56°, find :
(i) ∠DAE
(ii) ∠FEA
(iii) ∠EAC
(iv) ∠AEC

In parallelogram ABCD, E is the mid-point of side AB and CE bisects angle BCD. Prove that :
(i) AE = AD
(ii) DE bisects angle ADC
(iii) angle DEC is a right angle