Paragraph analysis:

• From the given paragraph, you want to analyse and find keywords that are there to remind you of your theorems, remember the theorem and go look for it in the diagram.
• ABCD is a cyclic quad:

Be on the look out for angles on same segment, which is not here because diagonal DB is not joined

Exterior angle of CQ: D2 is and exterior angle of the quad and is equal to B equal to x

Opposite angles of CQ: we will be on the look out for this one where necessary in questions

• AC is perpendicular to CB:

This gives us angles on semi-circle, making AB a diameter

• DC=CB: equal chords subtend equal angles, so AC bisects angle A, making A1 equal to A2
• AM is perpendicular to MC

This gives us triangle MDC of which we can use Pythagoras and trig ratios.

• Note: MC looks like a tangent but we are not told that it is, so we will not assume it to be a tangent

10.1.1

• To prove that MC is a tangent at C, either we prove that C2=A1 or angle MCA=x
• In triangle MDC and ABC:
• D2=B=x: exterior angle of a CQ
• C2=90-x: sum of angles in a triangle
• A2=90-x: sum of angles in a triangle
• C2=A2: both equal to 90-x
• But, A1=A2: equal chords subtend equal angles
• Therefore, A1=C2: both equal to A2
• MC is a tangent at C: converse Tanchord theorem

10.1.2

• Our job here is to find 3 equal angles in both the triangles
• In triangle ACB and CMD:
• M=C: both equals to 90
• D2=B: exterior angle of CQ
• C2=A2: third angle of a triangle, or proven above
• Therefore, triangle ACB is similar to triangle CMD: AAA

10.2.1

• Check where all the given sides of the proportion come from, they come from triangle CMD and ACB, but we have a problem with side AM from triangle AMC that we did not deal with.
• But triangle AMC is similar to CMD, M is common, C2=A1 proven above on Tanchord, D2=MCA by third angle of a triangle
• So, all these three triangles are similar and we can make similarity proportions from them
• This was not easy; it took me a lengthy time to find. 😊
• Check the answer on the board below, the part above the two lines was my scribble. Actual answer is the second part where I made proportions from triangle ABC, CMD, and AMC.
• It’s a matter of making valid proportions and play with them until you get to the required

10.2.2

• Once again, make possible proportions and play with them, inspect, invigilate, observe, etc.
• We can find sin x in triangle MDC and ABC using trig ratios
• And we can make proportions from the two triangles using similarity
• Then we inspect, invigilate and observe possibilities that can lead us to the required
• Check it out on the board bellow