In the diagram below; P and Q are midpoints of sides BC and AD respectively of the parallelogram ABCD. If side AB = diagonal BD; prove that the quadrilateral BPDQ is a rectangle.
Rectilinear Figures
1 Like

Question

In the diagram below; P and Q are midpoints of sides BC and AD respectively of the parallelogram ABCD. If side AB = diagonal BD; prove that the quadrilateral BPDQ is a rectangle.

KnowledgeBoat · Accepted Answer

Given: ABCD is a parallelogram. P and Q are midpoints of sides BC and AD, respectively. AB = BD In a parallelogram, opposite sides are equal and parallel. Thus, AB = CD and AD = BC. Since P and Q are midpoints, by the Midpoint Theorem, PQ || AB and PQ = AB. Similarly, BP = QD. In Δ QBD and Δ DPB, BD = BD (Common) QD = BP (midpoint property, QD = AD = BC = BP) ∠PBD = ∠QDB (alternate interior angles, since PQ || AB) Using SAS congruency criterion, Δ QBD ≅ Δ DPB Therefore, by corresponding parts of congruent triangles, QB = DP In quadrilateral BPDQ, QD = BP QB = DP Thus, opposite sides of BPDQ are equal. In Δ QAB and Δ DQB, AQ = QD (Q is mid-point of AD) AB = BD (Given) QB = QB (Common side) Using SSS congruency criterion, Δ QAB ≅ Δ DQB Therefore, by corresponding parts of congruent triangles, ∠AQB = ∠DQB ⇒ ∠AQB + ∠DQB = 180° (Linear pair of angles) ⇒ ∠AQB + ∠AQB = 180° ⇒ 2∠AQB = 180° ⇒ ∠AQB = ⇒ ∠AQB = ∠DQB = 90° Thus, BPDQ has opposite sides equal and two adjacent angles 90°. Since one angle of a parallelogram is 90°, all angles are 90°. Since BPDQ has opposite sides equal and all angles 90°, it is a rectangle. Hence, the quadrilateral BPDQ is a rectangle.

Mathematics

In the diagram below; P and Q are midpoints of sides BC and AD respectively of the parallelogram ABCD. If side AB = diagonal BD; prove that the quadrilateral BPDQ is a rectangle.

Rectilinear Figures

Related Questions

In a quadrilateral ABCD; angles A, B, C and D are in the ratio 3 : 2 : 1 : 4. Prove that AD is parallel to BC.

In a quadrilateral ABCD, AB = CD and ∠B = ∠C. Prove that:
(i) AC = DB,
(ii) AD is parallel to BC.

ABCD is a parallelogram. If AB = 2AD and P is the mid-point of CD; prove that : ∠APB = 90°.

Construct a parallelogram ABCD in which diagonal AC = 6.3 cm; diagonal BD = 7cm and the acute angles between the diagonals is 45°.

Mathematics

In the diagram below; P and Q are midpoints of sides BC and AD respectively of the parallelogram ABCD. If side AB = diagonal BD; prove that the quadrilateral BPDQ is a rectangle.

Rectilinear Figures

Related Questions

In a quadrilateral ABCD; angles A, B, C and D are in the ratio 3 : 2 : 1 : 4. Prove that AD is parallel to BC.

In a quadrilateral ABCD, AB = CD and ∠B = ∠C. Prove that:(i) AC = DB,(ii) AD is parallel to BC.

ABCD is a parallelogram. If AB = 2AD and P is the mid-point of CD; prove that : ∠APB = 90°.

Construct a parallelogram ABCD in which diagonal AC = 6.3 cm; diagonal BD = 7cm and the acute angles between the diagonals is 45°.

In a quadrilateral ABCD, AB = CD and ∠B = ∠C. Prove that:
(i) AC = DB,
(ii) AD is parallel to BC.