# Math Help - a question on properties of circle

1. ## a question on properties of circle

Referring to the attached pic, work out the length of OC - do not use trigonometric ratios stuff if possible.

2. Draw a tangent to the circle from C, say CT is the tangent and OT is the radius (so the lines CT and OT are perpendicular). There's a theorem which says that $CA\times CB = CT^2$, from which you can find CT. Then use Pythagoras.

3. Originally Posted by Opalg
Draw a tangent to the circle from C, say CT is the tangent and OT is the radius (so the lines CT and OT are perpendicular). There's a theorem which says that $CA\times CB = CT^2$, from which you can find CT. Then use Pythagoras.
produce 2 similar triangles?

4. Originally Posted by Opalg
Draw a tangent to the circle from C, say CT is the tangent and OT is the radius (so the lines CT and OT are perpendicular). There's a theorem which says that $CA\times CB = CT^2$, from which you can find CT. Then use Pythagoras.
but how do you know that TC is certainly a tangent of the circle???
actually if TC is produced from the original figure, angle ATC is does not appear to be a right angle.

5. Originally Posted by ukorov
but how do you know that TC is certainly a tangent of the circle???
actually if TC is produced from the original figure, angle ATC is does not appear to be a right angle.
This is the picture that I intended. The angle OTC is a right angle.

6. Originally Posted by Opalg
This is the picture that I intended. The angle OTC is a right angle.
oh i see but why is CA x CB = CT^2 ?

7. Originally Posted by ukorov
Referring to the attached pic, work out the length of OC - do not use trigonometric ratios stuff if possible.
Is it considered to be using trigonometric ratios
if you find the mid-point of the chord AB?

Since the chord length is given as 8, half would be 4.
If not then you have two 3-4-5 triangles inside the circle.
You then have sufficient information to complete the sides of a right triangle.
And from there, use pythagoras.

.

8. Originally Posted by aidan
Is it considered to be using trigonometric ratios
if you find the mid-point of the chord AB?

Since the chord length is given as 8, half would be 4.
If not then you have two 3-4-5 triangles inside the circle.
You then have sufficient information to complete the sides of a right triangle.
And from there, use pythagoras.

.
well it is pretty easy to solve using tri ratios but this question was extracted from a textbook chapter on properties of circle.

9. Originally Posted by ukorov
oh i see but why is CA x CB = CT^2 ?
Similar triangles. The triangles CTB and CAT are similar (angles are equal by the alternate segment theorem). Therefore $\frac{CA}{CT} = \frac{CT}{CB}.$