Dear forum members,
could someone please help me with this?
Find the distance of the point P(the intersection of the tangents) from the centre of the smaller circle.
The radius of the larger circle is 5, and the radius of the smaller is 3.That is all which is known.
Could someone please give me a hint of how to start doing this?
Thank you!
There's almost certainly a simple circle geometry theorem I've forgotten that will make this question easy and the solution elegant.
So in advance, here's a hamfisted solution outline (I'm assuming the two circles touch ...?) that might get you by until someone proves me right:
Let C1 be the centre of the r = 5cm circle (hereby called circle 1) and let T1 be where the tangent touches this circle. Let C2 be the centre of the r = 3 cm circle (hereby called circle 2) and let T2 be where the tangent touches this circle.
Use Pythag on triangle T1 C1 C2 to get T1 C2.
Use pythag on triangle T1 T2 C2 to get T1 T2. I can't be bothered working it out, so I'll call the value a.
Let the two tangents meet at P.
Let T2 P = b and C2 P = c.
Use Pythag on triangle C1 T1 P: 5^2 + (a + b)^2 = (8 + c)^2 ..... (1)
Use Pythag on triangle C2 T2 P: 3^2 + b^2 = c^2 .... (2)
Solve (1) and (2) simultaneously for c (remember that you know the value of a).
In your first method, I don't think Pythagoras is valid, because the diagram in my book does not state that the angle at C1 would be 90 degrees, and I don't know how valid an assumption as such would be. The place where the tangent touches the circles is perpedicular to the radius drawn to that point.
The only theoretical thing I was taught that could be applied to this, is that the angle at C1+ angle at P=180
Use Pythag on triangle T1 C1 C2 to get T1 C2.
True the angle at T1, is 90 degrees, but if I draw a hypotenuse from T1 to C2 then it will no longer be ninety. Then for this triangle T1 C1 C2 to become a right triangle, I would have to assume that the angle at C1 is 90 degrees, or am I just way off track here?
if I use the first method(since I don't know how to use the second one), I get the following equation
and I don't know how to go on solving it from here.
The trapezium diagram is attached. Do trig on the right-triangle C1 C2 A to get angle C1 C2 A. Angle C1 C2 T2 = 90 + angle C1 C2 A.
Angle T2 C2 P = 180 - angle C1 C2 T2. Then do trig on triangle T2 C2 P to get C2 P.
There a probably cleverer ways, but in the meantime see how you go.
Whoops, forgot tto attach the trapezium - see next post.