# Video: Pack 1 β’ Paper 3 β’ Question 24

Pack 1 β’ Paper 3 β’ Question 24

03:57

### Video Transcript

The points π΄, π΅, and πΆ all lie on the circumference of a circle with centre π. The line π΄πΆ passes through the point π. Prove that angle π΄π΅πΆ is right angled.

In order to solve this problem, we need to use our angle properties and our properties of circles. Firstly, we know that the length ππ΄ is equal to the length ππ΅ and also the length ππΆ as they all radii of the circle. As ππ΄ is equal to ππ΅, triangle ππ΄π΅ is isosceles. In the same way because ππ΅ is equal to ππΆ, ππ΅πΆ is isosceles.

As these triangles are isosceles, we know that angle ππ΅πΆ is equal to angle ππΆπ΅. We have defined this as π on our diagram. Likewise, angle ππ΄π΅ is equal to angle ππ΅π΄. We have defined this as π on the diagram. If we define angle π΄ππ΅ as the Greek letter π and angle π΅ππΆ as the Greek letter π, we know that π plus π equals 180 degrees as the sum of two angles on a straight line equals 180 degrees.

We also know that angles in a triangle add up to 180 degrees. If we consider the orange triangle ππ΄π΅, we can see that two π plus π equals 180. In the same way, in the pink triangle, two π plus π equals 180. Subtracting π from both sides of equation one gives us two π equals 180 minus π and dividing both sides of this equation by two gives us π is equal to 180 minus π divided by two. We can rearrange equation two in the same way. Two π is equal to 180 minus π and π is equal to 180 minus π divided by two.

The aim of our question is to prove that angle π΄π΅πΆ is right angled. Angle π΄π΅πΆ is equal to π plus π. Substituting in our expressions for π and π gives us 180 minus π divided by two plus 180 minus π divided by two. As we have a common denominator of two, we can rewrite this as 360 minus π minus π divided by two. Factorizing out a negative one from the last two terms on the top gives us 360 minus π plus π all divided by two.

We know that π plus π 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 π΄π΅πΆ 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 π΄πΆ is a diameter of a circle, angle π΄π·πΆ would be 90 degrees and angle π΄πΈπΆ would also be 90 degrees.

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