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?
Mid-point Theorem
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Question

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?

KnowledgeBoat · Accepted Answer

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 = AB (By mid-point theorem) ⇒ QM = PB ...(1) In △ABC, Since, P and Q are mid-points of AB and AC respectively. ⇒ PQ = 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.

Mathematics

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?

Mid-point Theorem

Related Questions

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :
DE = EF
AE = EF
∠DAE = ∠EFC
∠ADE = ∠ECF

In which of the following cases you will get 2XY = QR for the given figure?
(i) When PX = QX and PY = RY
(ii) When PX = QX and a + b = 180°
Only in case (i)
Only in case (ii)
In both the cases
None of these

In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :
3 LM
4 LM
6 LM
8 LM

In the given figure, A, B, C and D are mid-points of PQ, QR, RS and PS respectively. E, F, G and H are mid-points of AB, BC, CD and AD respectively. Which type of quadrilaterals are ABCD and EFGH?

Mathematics

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?

Mid-point Theorem

Related Questions

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :DE = EFAE = EF∠DAE = ∠EFC∠ADE = ∠ECF

In which of the following cases you will get 2XY = QR for the given figure?(i) When PX = QX and PY = RY(ii) When PX = QX and a + b = 180°Only in case (i)Only in case (ii)In both the casesNone of these

In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :3 LM4 LM6 LM8 LM

In the given figure, A, B, C and D are mid-points of PQ, QR, RS and PS respectively. E, F, G and H are mid-points of AB, BC, CD and AD respectively. Which type of quadrilaterals are ABCD and EFGH?

D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :
DE = EF
AE = EF
∠DAE = ∠EFC
∠ADE = ∠ECF

In which of the following cases you will get 2XY = QR for the given figure?
(i) When PX = QX and PY = RY
(ii) When PX = QX and a + b = 180°
Only in case (i)
Only in case (ii)
In both the cases
None of these

In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :
3 LM
4 LM
6 LM
8 LM