If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.
First, we need to calculate the lengths of the secant NM by adding x and 2.
NM=x+2
According to this theorem, the product of NM and PM is equal to the square of OM.
NM* PM =OM^2
Let's substitute NM with x+2, PM with x, and OM with sqrt(3) and solve the equation for x.
This is a quadratic equation. Let's solve it by completing the square. Note that all terms with x are on one side of the equation, so we do not need to rewrite it. In this equation, b=2. Using this information, we can calculate ( b2 )^2.
We conclude that the equation has two solutions.
x_1&=- 1+2=1
x_2&=- 1-2=- 3
On the diagram x represents the distance PM, so it cannot have a negative value. Therefore, x is equal to 1.
