Video: CBSE Class X • Pack 5 • 2014 • Question 11

CBSE Class X • Pack 5 • 2014 • Question 11

01:31

Video Transcript

Prove that the line segment joining the points of contact of two parallel tangents to a circle passes through the circle’s centre.

Let’s begin by sketching this out. Remember a diagram does not need to be to scale. But it should be roughly in proportion to help us decide how best to answer the question. Here, we have a circle with centre 𝑂 and two parallel tangents that meet the circle at the points 𝑃 and 𝑄. Let’s add in the lines joining the centre of the circle 𝑂 to the point where the tangents meet the circle at 𝑃 and 𝑄.

By their very definition, these are the radii of the circle. We know that the radii and the tangent form an angle of 90 degrees. Since the two tangents are parallel and the angle between both of the tangents and their radii is 90 degrees, this in turn means that the two radii formed by the line segments 𝑂𝑃 and 𝑂𝑄 must be parallel to one another.

We also know that the radii both pass through the centre of the circle 𝑂. This means they are part of the same line and the points 𝑃, 𝑂, and 𝑄 are collinear. We have shown that the two radii we added in are parallel and form part of the same line. This means they actually form a single straight line passing through the centre of the circle at 𝑂.

We have, therefore, proved that the line segment joining the points of contact of two parallel tangents to a circle passes through the circle centre.

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