In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.
Mid-point Theorem
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Question

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

KnowledgeBoat · Accepted Answer

By mid-point theorem, The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it. In △ABD, Since, P and Q are the mid-points of AB and BD respectively. PQ || AD ⇒ PQ = × AD .....(1) In △BCD, Since, R and Q are the mid-points of DC and BD respectively. QR || BC ⇒ QR = × BC ⇒ QR = × AD (∵ AD = BC) .....(2) In △ABC, Since, P and S are the mid-points of AB and AC respectively. PS || BC ⇒ PS = × BC ⇒ PS = × AD (∵ AD = BC) .....(3) In △ADC, Since, S and R are the mid-points of AC and DC respectively. SR || AD ⇒ SR = × AD .....(4) From eq.(1), (2), (3) and (4), we have: ⇒ PQ = SR = QR = PS Since, PQ || SR (Both are parallel to AD) and QR || PS (both are parallel to BC) ∴ PQRS is rhombus. Hence, proved that PQRS is a rhombus.

Mathematics

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

Mid-point Theorem

Related Questions

In the given figure, D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Prove that ADEF is a parallelogram.

If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.

In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that :
(i) ∠EFG = 90°
(ii) If GH || FE, then H bisects CB.

Mathematics

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

Mid-point Theorem

Related Questions

In the given figure, D, E, F are respectively the mid-points of the sides AB, BC and CA of △ABC. Prove that ADEF is a parallelogram.

If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle ABC, prove that △DEF is also an equilateral triangle.

In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that :(i) ∠EFG = 90°(ii) If GH || FE, then H bisects CB.

In the adjoining figure, ABCD is a kite in which AB = AD and CB = CD. If E, F, G are respectively the mid-points of AB, AD and CD, prove that :
(i) ∠EFG = 90°
(ii) If GH || FE, then H bisects CB.