Shown is a circle with centre
𝑂. 𝐴, 𝐵, and 𝐶 lie on the
circumference of the circle. Prove that 𝑦 is equal to two
𝑥. Do not use any circle theorems in
This is an example of a geometric
proof. In order to prove the statement
that 𝑦 is equal to two 𝑥, we need to use a series of known facts. We’ve been told that we’re not
allowed to use any circle theorems. Therefore, we will need to use the
angle facts we know.
Here, we have two isosceles
triangles. The first is triangle 𝐴𝑂𝐵 we
know that it’s an isosceles triangle because the sides 𝐴𝑂 and 𝑂𝐵 as shown are
the radii of the circle, which means they have to be of the same length. Since this is an isosceles
triangle, we know that angle 𝑂𝐴𝐵 and 𝑂𝐵𝐴 are the same size. We will call these 𝑎 for now.
Next, let’s use the fact that
angles in a triangle add to 180 degrees. We can find the remaining angle in
this triangle 𝐴𝑂𝐵 by subtracting the known angles from 180. Let’s call this angle 𝑐. 𝑐 is equal to 180 minus 𝑎 plus
𝑎, which can be simplified to 180 minus two 𝑎.
Next, we can repeat the process for
the other isosceles triangle 𝐵𝑂𝐶. We’ll call these two equal angles
𝑏 this time. Once again, since angles in a
triangle add to 180 degrees, the angle 𝐵𝑂𝐶 which we’ve now called 𝑑 is 180 minus
𝑏 plus 𝑏 or 180 minus two 𝑏.
The final fact that we need to use
is that angles around a point add to 360 degrees. This helps us find an expression
for the angle labelled 𝑦. 𝑦 is equal to 360 minus 𝑐 minus
𝑑. Remember we said that 𝑐 was equal
to 180 minus two 𝑎 and 𝑑 was equal to 180 minus two 𝑏. We can substitute these expressions
that we originally created for 𝑐 and 𝑑 into our equation for 𝑦.
Next, we’ll expand the brackets
really carefully. Remember since each bracket has a
minus sign in front of it, it’s technically being multiplied by negative one. That gives us 360 minus 180 plus
two 𝑎 minus 180 plus two 𝑏. Now, 360 minus 180 minus 180 is
zero. So our equation for 𝑦 is 𝑦 is
equal to two 𝑎 plus two 𝑏. And we can factorize this to get
two lots of 𝑎 plus 𝑏.
But let’s look back to our
diagram. Angle 𝑎 and 𝑏 combine to make
angle 𝑥. So 𝑥 is equal to 𝑎 plus 𝑏. In this case then, we will replace
𝑎 plus 𝑏 with 𝑥 and we get 𝑦 is equal to two 𝑥. That is what we’re being asked to
prove. So we are done.
Remember when we’re performing a
geometric proof, we must ensure that we explicitly write down all the angle rules
that we’ve used. Here that was rules about angles in
a triangle adding to 180 degrees, angles in an isosceles triangle, and angles around
a point adding to 360 degrees.