(1) In ∆ ABC, P, Q and R are the midpoints of sides AB, AC and BC respectively.
Seg AS=side BC. Prove that : PQRS is cyclic.
Answers
Quadrilateral PRQS is a cyclic quadrilateral.
Step-by-step explanation:
In ΔABC, R and Q are mid points of AB and CA respectively.
∴ RQ || BC (Mid point theorem)
Similarly, PQ || AB and PR || CA
In quadrilateral BPQR,
BP || RQ and PQ || BR (RQ || BC and PQ || AB)
∴ Quadrilateral BPQR is a parallelogram.
Similarly, quadrilateral ARPQ is a parallelogram.
∴ ∠A = ∠RPQ (Opposite sides of parallelogram are equal)
PR || AC and PC is the transversal,
∴ ∠BPR = ∠C (Corresponding angles)
∠SPQ = ∠SPR + ∠RPQ = ∠A + ∠C ...(1)
RQ || BC and BR is the transversal,
∴ ∠ARO = ∠B (Corresponding angles) ...(2)
In ΔABS, R is the mid point of AB and OR || BS.
∴ O is the mid point of AS (Converse of mid point theorem)
⇒ OA = OS
In ΔAOR and ΔSOR,
OA = OS (Proved)
∠AOR = ∠SOR (90°)
{∠ROS = ∠OSP (Alternate angles) & ∠AOR = ∠ROS = 90° (linear pair)}
OR = OR (Common)
∴ ΔAOR congruence ΔSOR (SAS congruence criterion)
⇒ ∠ARO = ∠SRO (CPCT)
⇒ ∠SRO = ∠B (Using (2))
In quadrilateral PRQS,
∠SRO + ∠SPQ = ∠B + ( ∠A + ∠C) = ∠A + ∠B + ∠C (Using (1))
⇒ ∠SRO + ∠SPQ = 180° ( ∠A + ∠B + ∠C = 180°)
Hence, quadrilateral PRQS is a cyclic quadrilateral.
To learn more...
1. In triangle ABC, pqr are themidpoint of side ab ,ac, bc.respectively seg as perpendicular bc prove that square pqrs is cyclic quadrilateral.
https://brainly.in/question/7250040
2. If P, Q and R are mid points of sides BC, CA and AB of a triangle ABC, and Ad is the perpendicular from A on BC, prove that P, Q , R and D are concyclic.
https://brainly.in/question/1134913
