Hi everyone!,

I am working through a paper, and have completed so to speak the majority of problems. I got stuck with another really tricky one. Please if you could, how can I prove "O" is the centre?

GIVEN:

1) AOIIQR

2) PA = AQ

3) PB = BT

First I had to prove ABIIQT, which I did. I just went by saying:

In triangle PQT:

PA=AQ Given

PB=BT Given

therefore AB disects two sides of triangle

therefore ABIIQT ...

NOW, prove "O" is the centre of the circle if PR is the center line of the circle.

All I do is stare at this...I have thought about the following however:

Since AO runs from the centre of PQ, parallel to the 3rd side (QR), does it not mean that AO will disect PR into two exact halves?, and since PR runs from one side of the circle to the other side of it, it would mean that PO is equal to OR...which tells me "O" is the centre?....I'm not sure if I'm at all close here...Please advise? Thank you all..

juriemagic
Sep 13, 2017

#1**+2 **

' Since AO runs from the centre of PQ, parallel to the 3rd side (QP), does it not mean that AO will disect PR into two exact halves ' ?

Yes it does, but this is what you are required to prove.

Show that the two triangles PAO and PQR are similar and then consider the ratio PA/PO.

Equate it with the corresponding ratio in the larger triangle and then use the property that PA = AQ.

Guest Sep 13, 2017

#2**+2 **

Hi guest,

thank you very much for nudging me into the right direction...I should be able to get this now...Thanx a lot!

juriemagic
Sep 13, 2017

#5**0 **

Hi Melody,

well, honestly....I went about the whole thing a different way...The reason I believe I cannot make use of ratio's, is because I do not have any values. I do not know how one can use ratio's if there are no values....so I did this:

In triangle PQR:

AOIIQR Given

PA=AQ Given

Therefore AO runs through the middel of triangle PQR

Therefore AO devides PR Midpoint theorem

Thus PO=OR

Since PR is the diameter, Given

PO must equal OR

Thus "O" is the middle of the circle.

Is this wrong?..Thanx for your assistance!!

juriemagic
Sep 13, 2017

#6**+2 **

Since AO is ll to QR then PR is a transversal that cuts parallel lines

Thus angle POA = angle PRQ

And angle QPR = angle APO

Thus, by Angle-Angle congruency....triangle QPR is similar to triangle APO

But similar triangles are similar in all respects.....

So........ PQ / PR = PA / PO which also implies that PA / PQ = PO / PR

But.....PA = AQ.....so.....PA = (1/2)PQ

Then....by substitution.......

(1/2) PQ / PQ = PO / PR

Simplifying.....we have that

(1/2) = PO / PR whicih implies that

PO = (1/2)PR

And since PR is the diameter of the circle and PO is (1/2) of PR.....O must be the center of the circle

CPhill
Sep 13, 2017

#8**0 **

CPhill,

I really wish I had the experience you guys have!!..It's good to see how it's solved, however, please if you do not mind....what in my solution makes it wrong?..I understand my approach was wrong, simply because you say it was, and you gave me the appropriate solution....but why is my approach wrong?..would you kindly spend just a little more time please, and just educate me..please...I need to understand this?..You guys are great, thank you for being out there!!

juriemagic
Sep 14, 2017

#9**0 **

Hi Chris,

I followed your logic right up till you said "And since PR is the diameter of the circle"

Why can you say that PR is the diameter of the circle ??

Maybe it is written somewhere and i am blind. I am not being sarcastic, this is always a distinct posibility :)

It is Red's fault! He stole my glasses!

Melody
Sep 14, 2017

#10**0 **

Hi Juriemagic,

Firstly you disect rats in a laboratory and you **bisect** intervals when you cut them in half :)

Yes I like your logic, although you would need to discuss similar triangles as Chris did.

I have the same problem with your logic as I do with Chris's, I can see that O bisects PR easily enough but I do not know why either of you has stated that PR is a diameter

I now see that the original guest answerer has made exactly the same 'assumption' ://

Melody
Sep 14, 2017

#12**0 **

haha...disects!!..oh my goodness!!!..so sorry, that was an honest boo boo...

Also, This exact question I found on the inernet in an exam question paper. The marks allocated for proving ABIIQT, was 2, and for proving the centre, also just 2. Sooo, I'm thinking going the similar triangle way is a lot of work for just 2 marks, HOWEVER, I do aknowledge that the approach given to me was far superior and most likely, the best approach.

Guys, you have all been a great help with this one, I really admire and love you all!!!

juriemagic
Sep 14, 2017

#14**0 **

LOL That is what I thought the first time I saw it but you did that boo boo a couple of times :)

Did you put your working in the middle of the question as I have asked further down the page?

Please be careful. It is very frustrating to put a lot of time into a question when the question is not presented properly. :/

Melody
Sep 14, 2017

#11**0 **

No assumption Melody.

Read the question.

NOW, prove "O" is the centre of the circle** **if PR is the centre line of the circle.

Guest Sep 14, 2017

#13**0 **

I thought that statement was a part of Juriemagic's working.

I did not think it was a part of the question!

------------------

This is the the question, along with the pic of course:

I am working through a paper, and have completed so to speak the majority of problems. I got stuck with another really tricky one. Please if you could, how can I prove "O" is the centre?

GIVEN:

1) AOIIQR

2) PA = AQ

3) PB = BT

No where does it say that PR is a diameter!

-------------------

NOW that I look at it with your eyes I can see that maybe Juriemagic put his working in the middle of his question.

All very confusing!

Melody
Sep 14, 2017

#15**0 **

Okay, let me clarify this...

The paper reads:

GIVEN:

1) AOIIQR

2) PA = AQ

3) PB = BT

Prove:

1) ABIIQT

2) O is the centre of the circle if PR is the centre line through the circle

3) BORT is a trapezium

Sorry if there was confusion, I did not realize giving it like it actually was on the paper would or could cause a mis-understanding. From my side, please accept my apologies.

juriemagic
Sep 14, 2017

#16**0 **

ok Juiemagic but you left out key components of the question and the other question I spent a lot of time on also had bits missing.

Last time I thought it was a badly written, incomplete question but maybe the question was just fine until you gave us the condensed version.

I certainly accept your appology but please do not give us half questions (or reworded questions) any more. :/

Melody
Sep 14, 2017