Video: CBSE Class X • Pack 5 • 2014 • Question 2

CBSE Class X • Pack 5 • 2014 • Question 2

01:46

Video Transcript

In the figure, 𝑄𝑅 is a common tangent to two circles which touch at point 𝑇. The tangent at 𝑇 meets 𝑄𝑅 at 𝑃. If 𝑃𝑇 is 3.8 centimetres long, find the length of 𝑄𝑅 in centimetres.

We’re given that the length of the tangent at the common point 𝑇 to the external point 𝑃 is 3.8 centimetres. And we are required to find the length of 𝑄𝑅; that’s the length of the common tangent between the two points of tangency.

The key to answering this question is the fact that the lengths of tangents drawn from an external point to a circle are equal. Taking the external point to be 𝑃 and ignoring the larger circle for a moment, we see that 𝑃𝑇 and 𝑃𝑅 are tangents drawn from the external point 𝑃 to the smaller circle. And so they are equal in length. And we know that the length of 𝑃𝑇 is 3.8 centimetres, so this is the length of 𝑃𝑅 also.

We do exactly the same to find the length of 𝑄𝑃. Ignoring the smaller circle now, we see that 𝑄𝑃 and 𝑃𝑇 are tangents drawn from the external point 𝑃 to the larger circle. And so by the theorem, they are equal in length. We’ve found the lengths of 𝑃𝑅 and 𝑄𝑃.

Remember, we’re looking for the length of 𝑄𝑅. That’s what we were asked to find in the question. Well, as 𝑃 lies on 𝑄𝑅, 𝑄𝑅 is just 𝑄𝑃 plus 𝑃𝑅. We can see this from the diagram. And so its length is 3.8 centimetres plus 3.8 centimetres, which is 7.6 centimetres.

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