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Q.

If the radii of two concentric circles are 4 cm and 5 cm, then find the length of chord of one circle which is tangent to the other circle.

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a

The length of the chord is 7 cm.

b

The length of the chord is 4 cm.

c

The length of the chord is 10 cm.

d

The length of the chord is 6 cm.

answer is D.

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Detailed Solution

Given that, two concentric circles of radii 4 cm and 5 cm.
Let us consider the following concentric circles with center A.
Question Image

Question Image

In the above figure, we can observe that, EF is the chord for the outer circle, and it is the tangent for the inner circle at point D.
Given that, radius of the inner circle = 4 cm = AD.
Also, radius of outer circle, AE = AF = 5 cm.
We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact. So,
∠ ADE = ∠ ADF = 90⁰ ………(1)
By applying, Pythagoras theorem in ∆ADE with the values AE = 5 cm, AD =4 cm, we get,
Hypotenuse² = Base² + Perpendicular²

AE 2 = AD 2 + ED 2 52 = 42 + ED 2 25 = 16 + ED 2 ED 2 = 9 ED = 9 ED = 3 … … … . (2)

By applying Pythagoras theorem in ∆ADF and substituting the values AF = 5 cm, AD = 4 cm, we get,

A F 2 = A D 2 + F D 2 52 = 42 + F D 2 25 = 16 + F D 2 F D 2 = 9 F D = 9 F D = 3 … … (3)

Add eq (2) and (3) we get,

E F = E D + F D E F = 3 + 3 E F = 6 cm

Hence, the length of the chord is 6 cm.
So, the correct option is (4).

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