In the given figure, △ABC is a scalene triangle in which ∠B = 90°. P is the mid-point of AB, PQ || BC and QM ⊥ BC. Which type of quadrilateral is PQMB?

Given,

∠B = 90° and PQ || BC

AB is the transversal.

⇒ ∠PBM = ∠APQ = 90° (Corresponding angles are equal)

⇒ ∠APQ + ∠BPQ = 180° (Linear pair)

⇒ 90° + ∠BPQ = 180°

⇒ ∠BPQ = 180° - 90°

⇒ ∠BPQ = 90°

Since, QM ⊥ BC

⇒ ∠QMB = 90°

In a quadrilateral PQMB,

⇒ ∠PBM + ∠BPQ + ∠QMB + ∠PQM = 360°

⇒ 90° + 90° + 90° + ∠PQM = 360°

⇒ 270° + ∠PQM = 360°

⇒ ∠PQM = 360° - 270°

⇒ ∠PQM = 90°.

All the angles of a quadrilateral = 90°

By converse of mid-point theorem,

A line drawn through the midpoint of one side of a triangle, and parallel to another side, will bisect the third side.

In △ABC,

Since, P is the mid-point of AB and PQ || BC, thus :

Q is mid-point of AC.

In △ABC,

Since, Q is the mid-point of AC and QM || AB (as both are perpendicular to BC), thus :

Mi si mid-point of BC.

In △ABC,

Since, Q and M are mid-points of AC and BC respectively.

⇒ QM = $\dfrac{1}{2}$ AB (By mid-point theorem)

⇒ QM = PB …(1)

In △ABC,

Since, P and Q are mid-points of AB and AC respectively.

⇒ PQ = $\dfrac{1}{2}$ BC (By mid-point theorem)

⇒ PQ = BM …(2)

From eq.(1) and (2), we have :

Since, opposite sides are equal and all the interior angles equals to 90°.

∴ PQMB is a rectangle.

Hence, PQMB is a rectangle.