Video: CBSE Class X • Pack 3 • 2016 • Question 23

CBSE Class X • Pack 3 • 2016 • Question 23


Video Transcript

Draw a circle of radius four centimetres. Construct two tangents to the circle inclined at an angle of 60 degrees to each other.

Now before we tackle this question, let’s just plan our approach. We’re gonna create a circle of radius four centimetres, and we’re gonna construct tangents to them that are 60 degrees to each other.

Now we know that two tangents from the same circle that meet at a point are equal in length. So this length is equal to this length. We also know that tangents are at right angles to the radii. Because they’re both radii, we know that these lengths are also equal.

This side is common to both those triangles, so we’ve got two congruent triangles. This angle here is gonna be equal to this angle here. In other words, each of those little angles is gonna be 30 degrees.

Now because angles in a triangle add up to 180 degrees, this angle here is gonna be 60 degrees, as is this one down here. So if I can create an angle of 60 degrees here, I can work out where these points need to be to create my tangents.

Right, let’s start our construction. Okay, put your compass point at zero and open them up to four centimetres. The more accurate you do this, the better. Now let’s draw the circle. Let’s mark our centre 𝑂, and we know the radius is four.

Now I’m gonna draw a line starting at 𝑂 and coming out here. So put the pencil point on 𝑂, bring the ruler in, swivel it around, and just draw that line here. So I’m gonna call this point 𝐴.

Now keeping the radius at four centimetres, using the centre of 𝐴, I’m gonna draw another circle. Now this distance and this distance and this distance are all the same, four centimetres. And if they’re all the same, that makes that an equilateral triangle. So each of these angles are 60 degrees. This must be the point where the top tangent touches the circle. Let’s call that 𝑊.

Likewise down here, let’s call that 𝑋. This is the point where the other tangent meets the circle. Again, I can do the same down here, and I would have 60 degrees here, here, and here. So the tangent to the circle is here, but how do I know where to draw that tangent too?

Well, this point out here, 𝐵, is also four centimetres away from here. Also, 𝑂𝐴𝐵 is a straight line, so if that is 60 degrees, this is 120 degrees, cause angles on a straight line add up to 180 degrees. Now if I draw in 𝑊 and 𝐵, we can see we’ve got an isosceles triangle. This is the same as this.

Now angles in a triangle sum to 180, so angle 𝐴𝑊𝐵 plus angle 𝐴𝐵𝑊 plus 120 degrees is equal to 180 degrees. Now we know that base angles in an isosceles triangle are equal, so we know that angle 𝐴𝑊𝐵 is equal to 30 degrees and that’s also equal to angle 𝐴𝐵𝑊. So we can see that the line that we drew here is at 90 degrees to this radius. That’s the definition of a tangent. We can do the same from 𝐵 to 𝑋. Now remember, this is 30 degrees; this is 30 degrees; so this angle here is 60 degrees, just like we wanted. Let’s just firm up those two tangents, and there’s our answer.

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