Geometry Question

Question

In the diagram AP  is secant to the circle at points A and B, and CP is secant to the circle at points C and D. The lines intersect outside of the circle at point P.

AB= 4

BP= (x-2)

CD= 7

DP= (x-3)

What is the value of x?

The value of x is _____.

Enter your answer as the number that correctly fills in the blank in the previous sentence, like this: 4

Answer 1

draw a diagram

in geometry if two secants intersect outside of a circle, you can form two products

the product of the (entire) secant segment and the external secant segment of one secant equals the product of the (entire) secant segment and the external secant segment of the other secant

keep in mind that secants extend indefinitely in both directions, that's why I am saying secant segments

therefore...

(x-2)(4+x-2)=(x-3)(7+x-3)

(x-2)(x+2)=(x-3)(x+4)

x^2-4=x^2+x-12

subtract x^2 from both sides

-4=x-12

x=12-4

x=8