Question

How would you solve this?

I got one of the equations to be 180 = 12y + 12x using tangent-secant theorem but I don't know what other equation to use to set up a system pls help

Answer 1

`x=6` and `y=9`

Explanation:

If a tangent segment and a secant segment are drawn to a circle (here larger circle) from an exterior point,

then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment. Hence

`18^2=12×(12+x+y)`

i.e. `12+x+y=324/12=27`

and `x+y=27-12=15`

Further, we have two tangents from same point to smaller circle and they should be equal. Therefore

`12+x=18` i.e. `x=6`

and hence `y=15-6=9`

Answer 2

`x=6` and `y=9`

Explanation:

If a tangent segment and a secant segment are drawn to a circle (here larger circle) from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment. Therefore

`18^2=12×(12+x+y)`

or `12+x+y=18^2/12=324/12=27` and

`x+y=27-12=15`

Further, for smaller circle from same external point we have two tangents and they should be equal i.e.

`12+x=18` or `x=18-12=6`

Hence `y=15-6=9`.