# Geometry of a Circle

• Oct 31st 2008, 05:23 AM
BG5965
Geometry of a Circle
Hi, I have 3 problems relating to the geometry of a circle. Also, can someone please explain the theorem relating to 'the tangent-chord angle being equal to an angle in the alternate segment'. I don't really understand that (the alternate segment part). Thanks!

1) In the figure, PQ is a tangent to the circle at A and OB is parallel to PQ. Find angle BAQ.
http://i301.photobucket.com/albums/nn74/BL5965/2.jpg

2) In the figure, the inscribed circle of triangle ABC meets the sides BC, CA and AB at P, Q and R respectively. If angle CAB is 50 degrees and angle ABC is 84 degrees, find all the angles of triangle PQR.
http://i301.photobucket.com/albums/nn74/BL5965/4.jpg

3) In the figure, O is the centre of the circle. A and B are two points of the circle such that OAB is an equilaterla triangle. OA is produced to C such that OA = AC. Find angle ABC and is CB tangent to the circle at B, giving a reason.
http://i301.photobucket.com/albums/nn74/BL5965/9.jpg

• Oct 31st 2008, 06:49 AM
earboth
Quote:

Originally Posted by BG5965
Hi, I have 3 problems relating to the geometry of a circle. Also, can someone please explain the theorem relating to 'the tangent-chord angle being equal to an angle in the alternate segment'. I don't really understand that (the alternate segment part). Thanks!

1) In the figure, PQ is a tangent to the circle at A and OB is parallel to PQ. Find angle BAQ.
http://i301.photobucket.com/albums/nn74/BL5965/2.jpg

...

I've modified your sketch. Since ACBO is a square the angle $\angle(QAB) = 45^\circ$

With my figure you easily can prove that the tangent-chord angle is have as large as the center angle $\angle(AOB)$.