Circle theorems - Find the radius?

Question

Take a look at the picture below, it is the diagram required to solve the question.
http://img231.imageshack.us/img231/3233/questiongw8.jpg
The question is:
AX = 6cm
BX = 10cm
The larger circle has a radius of 7cm.
What is the radius of the smaller circle?
State clearly any geometrical properties you use.

Answer 1

The easiest way to solve this problem is to use similar triangles, and create them from the points o1xa and o2xb.
As you can deduce that each triangle has two angles that are the same, they are therefore similar.

Because the triangles are similar, you can set up a proportion:
10 is to 7 as 6 is to the radius of a
So, 10/7 = 6/r
Cross multiply:
10r = 42
r = 42/10
r = 4.2

Answer 2

7/10 = x / 6 where x is the radius of the smaller circle. We are using the similar triangles principles.

So, x = 42/10 = 4.2 cms

Answer 3

Hi,

7..........r
---..=..----
10.......6

Cross-multiplying, 10r = 42

r = 4.2

That is your radius.

The reason this works is that there are congruent vertical anges at X, and tangents to a circle from the same point are equal in length. That means the 2 segments tangent to the left circle are congruent to each other and form an isosceles triangle. The 2 segments tangent to the right circle are also congruent to each other and form an isosceles triangle. Since they have their third angles congruent, these base angles of the triangles are also congruent. This allows you to conclude that these triangles are similar and the radii of their circles have the same ratio. therefore you can use the proportion as shown above.

I hope that helps!! :-)

Answer 4

The triangles O1AX and O2BX are similar because
(1) angle O1AX =angle O2BX =90 degrees because they are formed by a radius drawn to the point of tangency, and
(2) angle O1XA = angle O2XB because they each = 1/2 of the equal vertical angles formed by the intersecting tangents.

Hence O1A/ O2B= AX/BX
O1A =O2B*AX/BX = 7*6/10 = 4.2
r = 4.2

Answer 5

ya man you gotta take the ratio...it's the best thing to do...
first thing is..
1/ 10/6 = 7/x (x is the radius of the smaller circle)
2/ 10x = 42 Multiplication Property
3/ X = 42/10 Division Property
4/ X = 4.2 ty later dude hope this help!!

Answer 6

i'm going to assume that the circle you defined relies upon on the inspiration (0,0). eco-friendly's theorem relates the line necessary around the boundary dC of a area of the (x,y) plane to an necessary over the area C enclosed by skill of that boundary: LineInt(L*dx - M*dy, around dC) = AreaInt(dM/dx - dL/dy, over C) the place for this reason dM/dx and dL/dy are partial derivatives. we've L = y^2 and M = x^2, so as that dM/dx = 2*x dL/dy = 2*y and so we'd desire to evaluate AreaInt(2*x - 2*y, over a disk of radius 7). Int(-7

Answer 7

TRIANGLES OXB &OXA ARE SIMILAR
R/7 = 6/10
R=4.2