Circle Geometry (1 Viewer)

[Kutay]

Hey i was wondering if someone could tell me how to solve this questions.

Q1. TA, TB are tangents to the cirlce from T. Angle ATB = 50
(I) find Angle APB, give reasons



Q2. QR, PQ, PS, PS are tangents to the circle at D, C, B, A respectivily.
if Angle QRS = 70 and Angle PQr = 84 find Angle CBA, give reasons

I HAVE THE DIAGRAMS BUT ARE TOO BIG TO UPLOAD!!!!!!!!!!1
can someone reply so i can send them an email with it in ?

 

[pLuvia]

Q1. TA, TB are tangents to the cirlce from T. Angle ATB = 50
(I) find Angle APB, give reasons

Where's P?? do you O as in the centre of the circle?

 

[LostAuzzie]

You can email the diagram to me if you like, lost_auzzie@hotmail.com. Ill try and solve it with my own diagram in the mean time

 

[word.]

Hey i was wondering if someone could tell me how to solve this questions.

Q1. TA, TB are tangents to the cirlce from T. Angle ATB = 50
(I) find Angle APB, give reasons



Q2. QR, PQ, PS, PS are tangents to the circle at D, C, B, A respectivily.
if Angle QRS = 70 and Angle PQr = 84 find Angle CBA, give reasons

I HAVE THE DIAGRAMS BUT ARE TOO BIG TO UPLOAD!!!!!!!!!!1
can someone reply so i can send them an email with it in ?

If you can fix the questions then diagrams aren't needed...

 

[Kutay]

You can email the diagram to me if you like, lost_auzzie@hotmail.com. Ill try and solve it with my own diagram in the mean time

Sent check your email thankyou Kutay

 

[LostAuzzie]

Ill start with Q1
AT = TB (Tangents from a point)
Therefore ATB is isosceles
Angle TAB = Angle ABT (Base angles of isos triangle)
Angle TAB + Angle ABT = 180 - 50 (Angle sum of triangle)
= 130
Therefore Angle TAB = Angle ABT = 65
Now Angle PAB + Angle PBA = 65 (The best reason I can give for that is that the sum of these two angles is always the same and the limit as P approaches one of the points A or B is the size of that angle)

Angle APB = 180 - 65 (Angle sum of triangle)
= 115

Please correct me if there is something wrong here
EDIT: Attached is the diagram for this question

 

[Jago]

"Now Angle PAB + Angle PBA = 65 (The best reason I can give for that is that the sum of these two angles is always the same and the limit as P approaches one of the points A or B is the size of that angle)"

i don't really understand this part...

 

[LostAuzzie]

Now Q2:

Angle QCA = Angle CBA (Angle between tangent and chord)
Angle RAC = Angle CBA (Similarly)
Therefore Angle QCA = Angle RAC
Angle QCA + Angle CAR + Angle ARQ + Angle RQC = 360 (Angle sum of quad)
2Angle CBA + 84 + 70 = 360
2Angle CBA = 206
Angle CBA = 103

Once Again please correct me if something is wrong here
Also Below is a diagram for this question

 

[LostAuzzie]

"Now Angle PAB + Angle PBA = 65 (The best reason I can give for that is that the sum of these two angles is always the same and the limit as P approaches one of the points A or B is the size of that angle)"

i don't really understand this part...

The sum of the two angles is always the same because the third angle is always the same due to angles in same segment rule.
I dont believe my explanation is an actual rule as at that pint I got stuch for an official rule explaining it but...
As P moves closer to one of the points A or B the angle at that point gets larger while the angle at the other point gets smaller.
There is a limit to this, reached when P reaches the point either A or B, Here the larger angle is 65 while the shorter angle is 0 (its not a triangle now but thats why its a limit)
And hence, the sum of the two angles is 65
However this is not a circle geometry rule so Im hoping someone else will correct me with the circle geometry rule that proves this.

 

[insert-username]

LostAuzzie, I think you're right for the first question. You probably should word it like this, though:

Q1. TA, TB are tangents to the cirlce from T. Angle ATB = 50
(I) find Angle APB, give reasons

AT = AB (tangents from a point).

Therefore angle TAB = ABT = 65 degrees.

Angle PBA + Angle PBT = 65 (adjacent angles)

But Angle PBT = Angle PAB (angles in alternate segments)

Therefore Angle PBA + Angle PAB = 65

Therefore Angle APB = 180 - 65 (angle sum triangle)

Therefore Angle APB = 115 degrees.


I_F

 

[LostAuzzie]

Thanks for that, I knew there was some reason why the sum of those two angles was 65.

 

[word.]

Algebraically:
Using LostAuzzie's diagram:

let angle BAP = x, angle ABP = y

then angle PBT = x (angles in alt. segment)
also angle PAT = y (similarly)

angle APB = 180 - y - x (angle sum of triangle APB)
= 180 - (x + y)

also angle BAP + angle PAT = x + y = 65 degrees (base angle of isosceles triangle BAT)

so angle APB = 180 - 65 = 115 degrees

 

[haboozin]

oh man i just did a question with that diagram in it (Q2)..

you guys can have a go too... im had no chance with III.

 

[insert-username]

haboozin: What's the result of part (b)?


I_F

 

[haboozin]

haboozin: What's the result of part (b)?


I_F

part b is below part c...

or u mean u cant get the answer?
even if you cant get it, the inequality is there for u to use.

 

[insert-username]

Sorry, didn't see part b there under it. I'll have another look at the question.


I_F

 

[LostAuzzie]

The inequality is chopped off at the bottom all I see is 'given that PR - PQ = QR show that' and then its cut off.