The points 𝐴, 𝐵, and 𝐶 all lie
on the circumference of a circle with centre 𝑂. The line 𝐴𝐶 passes through the
point 𝑂. Prove that angle 𝐴𝐵𝐶 is right
In order to solve this problem, we
need to use our angle properties and our properties of circles. Firstly, we know that the length
𝑂𝐴 is equal to the length 𝑂𝐵 and also the length 𝑂𝐶 as they all radii of the
circle. As 𝑂𝐴 is equal to 𝑂𝐵, triangle
𝑂𝐴𝐵 is isosceles. In the same way because 𝑂𝐵 is
equal to 𝑂𝐶, 𝑂𝐵𝐶 is isosceles.
As these triangles are isosceles,
we know that angle 𝑂𝐵𝐶 is equal to angle 𝑂𝐶𝐵. We have defined this as 𝑎 on our
diagram. Likewise, angle 𝑂𝐴𝐵 is equal to
angle 𝑂𝐵𝐴. We have defined this as 𝑏 on the
diagram. If we define angle 𝐴𝑂𝐵 as the
Greek letter 𝜙 and angle 𝐵𝑂𝐶 as the Greek letter 𝜃, we know that 𝜙 plus 𝜃
equals 180 degrees as the sum of two angles on a straight line equals 180
We also know that angles in a
triangle add up to 180 degrees. If we consider the orange triangle
𝑂𝐴𝐵, we can see that two 𝑏 plus 𝜙 equals 180. In the same way, in the pink
triangle, two 𝑎 plus 𝜃 equals 180. Subtracting 𝜙 from both sides of
equation one gives us two 𝑏 equals 180 minus 𝜙 and dividing both sides of this
equation by two gives us 𝑏 is equal to 180 minus 𝜙 divided by two. We can rearrange equation two in
the same way. Two 𝑎 is equal to 180 minus 𝜃 and
𝑎 is equal to 180 minus 𝜃 divided by two.
The aim of our question is to prove
that angle 𝐴𝐵𝐶 is right angled. Angle 𝐴𝐵𝐶 is equal to 𝑎 plus
𝑏. Substituting in our expressions for
𝑎 and 𝑏 gives us 180 minus 𝜃 divided by two plus 180 minus 𝜙 divided by two. As we have a common denominator of
two, we can rewrite this as 360 minus 𝜃 minus 𝜙 divided by two. Factorizing out a negative one from
the last two terms on the top gives us 360 minus 𝜃 plus 𝜙 all divided by two.
We know that 𝜃 plus 𝜙 is equal to
180 degrees as they are two angles on a straight line. We, therefore, have 360 minus 180
divided by two. 360 minus 180 is 180. Dividing this by two gives us an
answer of 90 degrees. We have, therefore, proved that
angle 𝐴𝐵𝐶 is a right angle as it is equal to 90 degrees.
This is actually the proof of one
of our circle theorems. Any angle in a semicircle or from a
diameter is equal to 90 degrees. As 𝐴𝐶 is a diameter of a circle,
angle 𝐴𝐷𝐶 would be 90 degrees and angle 𝐴𝐸𝐶 would also be 90 degrees.