Plane geometry is one of the top killer questions in A-Math as there are many things to prove in addition to the many lines, circles, triangles, angles ...
Read about the post I wrote earlier What you need to know to do well in Plane Geometry Part 1
I must admit this question almost drives me to insanity ;-) but luckily I saw the light when I was on the verge of sane and insanity ...
Question: Prove that a circle can be drawn passing through the points T,P,X and Q. [3 marks]
Let me know how you handle this question in the comments section. For those who wants the step by step solution, drop me a comment as well. I highly recommend that you think about this question first before asking for solution.
There are 2 strategies I have used to help me in this Plane Geometry question. Do you want to know? so that you can learn and use them in your own question.
Hint: You must use one of the properties of circles.
Update: Video solution Here
Ai Ling Ong
Hi, I'm Ai Ling Ong. I enjoy coaching students who have challenges with understanding and scoring in 'O' Level A-Maths and E-Maths. I develop Math strategies, sometimes ridiculous ideas to help students in understanding abstract concepts the fast and memorable way. I write this blog to share with you the stuff I teach in my class, the common mistakes my students made, the 'way' to think, analyze... If you have found this blog post useful, please share it with your friends. I will really appreciate it! :)
I really donno, haha, so can you show the solutions?
As PT and PQ are of equal length, PTQ is an isosceles triangle, thus the circle can be drawn passing through these points by circumscribing the triangle....
Hi Always Lovely,
This one is indeed tough...yet to come out with my answers..
Rgds
Sean
Hmm this question is difficult, don't even know where to start. Mind revealing the solution?
to add on my previous reply:QXPT is a cyclic quad because the opp <s = 180degrees...
hmm if its wrong may i have the solution :)
<QXT = <QPT
<XQP = <XTP
<s in the same segment
sorry for double reply! :)
@anjali: As this is a 3 marks proving question, you won't get the full marks by stating the property of triangle PTQ. Nevertheless. good attempt :)
@Joscelin: Joscelin, you can't begin with the circle already passes through the 4 points and make the conclusions as above. You have to prove your conclusions to be correct hence the 4 points pass through the circle. This is also the challenginng part of many students for a proving question but you got the main concept correct. Begin the proving with the fact that angles in the same segment are equal.
I have updated the solution here => https://www.singaporeolevelmaths.com/2008/10/11/video-solution-of-plane-geometry-question-plusstrategy-to-handle-angles/
Angles XPT = Angles XQT = 90(tangent perpendicular to radius)
Using the property, Angles in a semicircle, a circle passing thru T, P, X and Q can be constructed with XT as a diameter.
angle xtq=rqz(corresponding angles)
but,angle rqz=rpq(angles in the alternate segment)
.:.angle xtq=rpq
.:.xpq=rpq