Q) Two concentric circle are of radii 4 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Ans:

Step 1: Let’s draw a diagram for our better understanding of the question:

Here, we have 2 concentric circles, both having O as centre.

Radius of smaller circle is, OP = 3 cm

and radius of larger circle, OB = 4 cm

Here, AB is the chord of larger circle and it is also tangent of smaller circle.

Step 2: By circle’s identity, A radius drawn on a tangent is perpendicular

So, for smaller circle, where OP is radius and AB is the tangent

∴  ∠ OPB = 90⁰

and ∴  Δ OPB is a right angled triangle.

Step 3: By circle’s identity, A perpendicular line drawn on a chord bisects it

So for larger circle, OB is the perpendicular radius on chord AB

∴  AP = PB

∴ AB = AP + PB = 2 PB

Step 4: Next, By Pythagoras theorem in right angled triangle Δ OPB,

OB² = OP² + PB²

∴ (4) ² = (3) ² + PB²

∴ 16 = 9 + PB²

∴ PB² = 16 – 9 = 7

∴ PB = √ 7

Step 5: ∵  AB = 2 PB           (from step 3)

∴ AB = 2 (√ 7) = 2  √ 7

Therefore, length of the chord of larger circle is 2√7 cm.

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