Pz solve the problem with logic with correct explaination
amit arya , 8 Years ago
Grade 12th pass
6 grade maths> Pz solve the problem with logic with corr...
1 Answers
Swati
Last Activity: 4 Years ago
Given : In ΔABC, P, Q and R are the mid points of sides BC, CA and AB respectively. AD ⊥ BC.
To prove : PQRD is a cyclic quardilateral.
Proof :
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)
∠DPQ = ∠DPR + ∠RPQ = ∠A + ∠C ...(1)
RQ || BC and BR is the transversal,
∴ ∠ARO = ∠B (Corresponding angles) ...(2)
In ΔABD, R is the mid point of AB and OR || BD.
∴ O is the mid point of AD (Converse of mid point theorem)
⇒ OA = OD
In ΔAOR and ΔDOR,
OA = OD (Proved)
∠AOR = ∠DOR (90°) {∠ROD = ∠ODP (Alternate angles) & ∠AOR = ∠ROD = 90° (linear pair)}
OR = OR (Common)
∴ ΔAOR congruence ΔDOR (SAS congruence criterion)
⇒ ∠ARO = ∠DRO (CPCT)
⇒ ∠DRO = ∠B (Using (2))
In quadrilateral PRQD,
∠DRO + ∠DPQ = ∠B + ( ∠A + ∠C) = ∠A + ∠B + ∠C (Using (1))
⇒ ∠DRO + ∠DPQ = 180° ( ∠A + ∠B + ∠C = 180°)
Hence, quadrilateral PRQD is a cyclic quadrilateral.
Provide a better Answer & Earn Cool Goodies
Enter text here...
LIVE ONLINE CLASSES
Prepraring for the competition made easy just by live online class.
Full Live Access
Study Material
Live Doubts Solving
Daily Class Assignments
Ask a Doubt
Get your questions answered by the expert for free
Enter text here...
Other Related Questions on 6 grade maths
Find the smallest number which when diminished by 3 is divisible by 21, 28, 36 and 45.
6 grade maths
1 Answer Available
Last Activity: 2 Years ago
Find the LCM of 12, 15, 20 27 by the division method.
6 grade maths
1 Answer Available
Last Activity: 2 Years ago
Fill in the blank: Collection of information is called____
6 grade maths
2 Answers Available
Last Activity: 2 Years ago
Which of the following is a composite number ?
6 grade maths
1 Answer Available
Last Activity: 2 Years ago