# circle problem..

• Feb 28th 2010, 07:11 AM
snigdha
circle problem..
If O is the centre of the circle. Prove that :
< x + < y = < z
• Mar 1st 2010, 01:38 AM
Hello snigdha
Quote:

Originally Posted by snigdha
If O is the centre of the circle. Prove that :
< x + < y = < z

I have added the letters A, B and C to your diagram - see the attached.

Then
$\angle QAR = \angle QBR = \tfrac12z$ (angle at centre)

$\Rightarrow \angle PAC = \angle PBC = 180^o-\tfrac12z$ (angles on a straight line)

$\Rightarrow x + y +(180^o-\tfrac12z)+(180^o-\tfrac12z)=360^o$ (angle sum of quadrilateral ACBP)

$\Rightarrow x + y+360^o-z =360^o$

$\Rightarrow x + y = z$