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
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
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.