Question

QR is a chord of a circle and PQR is diameter of the circle.OD is perpendicular upon QR.If OD = 4 cm than what is the value of PQ.​

Answer 1

Answer:

QR is a chord of the circle with centre at O and POR is a diamter of it. OD is perpendicular to QR. If OD= 4 cm , then the length of PQ is. ... ∴ the length of PQ= 8 cm.

Step-by-step explanation:

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Answer 2

Answer : 8cm

Solution :

OD⊥QR∴∠ODR= 1 right angle

Again POR is a diameter of the circle with centre at O.

∴∠PQR is an angle is semicircle.

∴∠PQR =1 right angle.

∴OD∣∣PQand∠PRQ is common to both ΔPQRandΔODR

∴ΔPQRandΔODR are similar triangle.

∴ODPQ=ORPRor,4PQ=OR2QR[∴PR=2OR]or,4PQ=12ork,PQ=8

∴ the length of PQ= 8 cm ∴ (c) is correct.

Aliter : POR is a diamter of the circle with center at O.∠PQR is an angle in semicircle.

∴PQR= right angle or 90∘

∴ΔPQR is a right- angled triangle of which PR is the hypotenuse.

∴PR2=PQ2+Qr2----(1) [ by Pythagoras' theorem]

Again OD⊥QR, where is a chor of the circle.

∴D is the mid-point of QR,∴DR=12QR

∴ΔODR is a right-angled triange of which OR is the hypotenus.

∴OR2=OD2+DR2

or,(12PR)2=42+(12QR)2[∴QR=12PRandDRandOD=4cm]

or,PR24=16+QR24or,PR2=64+QR2 -----(2)

Now, from (1) and (2) we get,

PQ2+QR2=64+QR2

or,PQ2=64or,PQ64−−√=8

∴ the length of PQ =8 cm.

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