Two circles

[jimi]

I'm having trouble with this problem and would appreciate any help:

'Two circles of radii 5 cm and 3 cm touch externally at P. A common tangent to the circles touches the larger at S and the smaller at T. The tangent at P meets ST at X. Calculate the lengths of ST, PX and find the radius of the circle which passes through X and the centres of the two circles.'

Thanks for any help,

jimi.

 

[wjm11]

'Two circles of radii 5 cm and 3 cm touch externally at P. A common tangent to the circles touches the larger at S and the smaller at T. The tangent at P meets ST at X. Calculate the lengths of ST, PX and find the radius of the circle which passes through X and the centres of the two circles.'

Hi, Jimi,

Hint: extend line ST and the line thru the centers until they intersect. You will find two similar, right triangles have been created. You know the lengths of their bases and the difference in the lengths of their hypotenuses. Therefore, you can solve the triangles. This will give you the length of ST. You can also use trig, now, to solve the angles, which will allow you to solve for PX. Alternatively, you can still use similar triangles to solve for PX.

Let us know if you need more help.

 

[jimi]

Thanks wjm, I took your hint and solved it. Appreciate your help.

 

[Denis]

Thanks wjm, I took your hint and solved it. Appreciate your help.

What did you get jimi?

ST = 2sqrt(15) ?

PX = 15tan(q) where q=arcsin(1/4) ; approx 3.8729833... ?

and radius of circle = 4 ?

Let R = larger radius, r = smaller radius;
then length of line from center of larger circle to point of intersection
of line through the centers and tangent line is: R(R + r) / (R - r);
you may find that useful in other similar problems.

In your problem, that would be: 5(5 + 3) / (5 - 3) = 40 / 2 = 20.

Did you notice that the center of the 3rd circle was on the line going through
the centers of the other 2 circles, exactly 1 unit from P ?

 

[jimi]

Denis, thanks for replying.

'ST = 2sqrt(15) ?

PX = 15tan(q) where q=arcsin(1/4) ; approx 3.8729833... ?

and radius of circle = 4 ?

Let R = larger radius, r = smaller radius;
then length of line from center of larger circle to point of intersection
of line through the centers and tangent line is: R(R + r) / (R - r);
you may find that useful in other similar problems.

In your problem, that would be: 5(5 + 3) / (5 - 3) = 40 / 2 = 20.

Did you notice that the center of the 3rd circle was on the line going through
the centers of the other 2 circles, exactly 1 unit from P ?'


Yes, I agree with you:
ST = 2sqrt(15)
PX = sqrt(15)
radius of third circle = 4 cm

I didn't know about the standard results, R(R + r) / (R - r) or the location of the centre of the third circle. Thanks for enlightening me.
This question has opened my eyes to the many properties of circles and tangents. I had no idea there were so many.
Thanks to both you and wjm, much appreciated.

jimi.