Final answer:
To find the length of LM, we equate the given expressions for JL and ML since they represent equal tangents to circle K from a common point. After solving for x, we calculate LM to be 22 units.
Step-by-step explanation:
The statement of the problem involves a geometric configuration with a circle and tangents, leading to properties related to the lengths of tangent segments. To solve this, we use the fact that the tangent segments from a common external point to a circle are equal in length. Given that JL = 3x + 10 and ML = 7x - 6, and that both are tangents from the common point L to the circle K, we can set these expressions equal to each other because the lengths must be the same.
3x + 10 = 7x - 6
Solving for x gives us:
10 + 6 = 7x - 3x
16 = 4x
x = 4
Once we have the value of x, we can substitute it back into the expression for ML to find its length:
ML = 7(4) - 6 = 28 - 6 = 22
Therefore, the length of LM is 22 units.