Clicle Geo Q Debate (1 Viewer)

[NickP101]

Hey all in my half yearly there was a circle geo question which i just wanted to ask about. Angles in the same segment of a circle are equal, true?

Well if that is true, then assuming i have the diagram below:

Can i say that ABCD is a cyclic quad, because the angles shaded in black are =??

 

[Riviet]

Yes and yes. So go and claim your deserved marks back, if that's why you wanted re-assurance from this thread.

 

[NickP101]

Haha yeah that was why i posted this thread lol thanks!!!

 

[Trebla]

Assuming that A, B, C and D lie on the circumference of the circle then yeah...........

 

[NickP101]

Well thats what im trying to prove, by saying those angles are equal, can i saw that ABCD lie on the circumference of a circle, therefore making the four points a cyclic quad?

 

[Riviet]

You can say, since acute angles B and C are subtended by a common interval AD as well as equal, then A,B,C,D are concyclic.

 

[SoulSearcher]

I can give you a general proof for that, but the short answer is yes.

 

[shsshs]

as long as you explained it properly.. maybe lyk say (converse of angles subtended by equal chords/arcs)

 

[NickP101]

SoulSearcher do u think you could post that general solution, or maybe set me on the right path for proving it? My teacher said if i want the marks i have to prove it, im having trouble doing that so far.

 

[bboyelement]

SoulSearcher do u think you could post that general solution, or maybe set me on the right path for proving it? My teacher said if i want the marks i have to prove it, im having trouble doing that so far.

is your teacher even a maths teacher ... check your maths book ... there should be diagram some where showing equal angles in the same segment

 

[NickP101]

Thats not what im trying to prove though, im trying to prove that if the situation in diagram two occurs (in my first post) you can say that the four points are a cyclic quad.

 

[bboyelement]

Thats not what im trying to prove though, im trying to prove that if the situation in diagram two occurs (in my first post) you can say that the four points are a cyclic quad.

angle B = angle C
(equal angles in the same segment)
hence points A,B,C,D are concyclic and ABCD is a cyclic quad.

is that wat you want bcause from reading all the replies above ppl have been saying the exact thing

 

[rama_v]

Thats not what im trying to prove though, im trying to prove that if the situation in diagram two occurs (in my first post) you can say that the four points are a cyclic quad.

But thats the converse, and it is true. For example, if teh base angles are equal in a triangle, then the triangle is isosceles, and therefore if the triangle is isosceles, then the base angles are equal. Its the same situation here.

Besides in the HSC there are a list of theorems that you can take for granted and do not have to prove when using them, unless specifically asked to do so. This is one of them.

The sylabus states:

2.9.

3 Unit students will be expected to be able to prove any of the following
results using properties obtained in 2.3 or 2.8.
.
.
.
If an interval subtends equal angles at two points on the same side of it
then the end points of the interval and the two points are concyclic.
.
.
2.10
In applications to problems, any of the definitions given or results obtained
in 2.2, 2.3, 2.7, 2.8 or 2.9 may be used without proof provided a specific
reference is made to each result so used. If a proof is required for any of
the results in 2.9 then this will be clearly indicated.

http://www.boardofstudies.nsw.edu.au/syllabus_hsc/pdf_doc/maths23u_syl.pdf - Page 28

In other words, your teacher is incorrect, and you can fight for your marks.

 

[NickP101]

Haha yeah thats what i said to my teacher, he doesnt believe its true, so he wants me to prove it. I dont where i would start in proving it.

 

[insert-username]

You can show him the syllabus point that rama v posted. Print out that page from the syllabus - it's exactly what your question has. That should be more than enough to prove it.


I_F

 

[NickP101]

Haha Rama you champion, thats awsome, thanks mate!!!

 

[rama_v]

Haha Rama you champion, thats awsome, thanks mate!!!

Glad to help

 

[SoulSearcher]

General proof if you're interested.

Prove that the points A, B, P and Q are cyclic (shown on diagram) are concyclic, given that angle APB = angle AQB
To do this, draw a circle through the points A, B and P, and then extend the point Q so that it meets the circle at X.

angle AXB = angle APB (angles on the same arc AB)
but angle APB = angle AQB (given)
therefore angle AXB = angle AQB
so that OB ll XB (corresponding angles are equal)
But since QB and XB intersect at B, and since they are parallel lines, they are the same line, therefore Q and X coincide, and thus Q lies on the circle.