For a given point and a circle, the product of the lengths of the segments created by the point and the points of intersection with the circle is constant for any line.
x=25.8, y≈ 12.4
Practice makes perfect
Consider the given diagram.
We will find the values of x and y one at a time.
Finding x
Let's start by focusing on the secants. There are two points of intersection along each line through the point and the circle. We can think about these points of intersection as creating two different segments between the point and the circle — the point to the first point of intersection, and the point to the second point of intersection.
The product of the lengths of the segments created by the point and the points of intersection is constant for any line.
Therefore, in our diagram, the products of the secants and their outer segments are equal. With this, we can write and solve an equation in terms of x.
Let's now focus on the tangent and one of the secants. Again, we will use the fact that the product of the lengths of the two segments between the point to the points of intersection with the circle is constant along any line through the point and circle.
In the case of tangents there is only one point of intersection, but we still need to consider two segments.
Therefore, the product of the secant and its outer segment equals the square of the tangent. Let's write and solve an equation!
