Question

what is the length of line segment pq?
○ 4 units
○ 5 units
○ 6 units
○ 9 units
in the diagram: line segment nm = 6, line segment nq = 4, line segment pq = x

Explanation:

Step1: Recall secant-tangent theorem

For a point outside a circle, if a tangent segment and a secant segment are drawn to the circle, then the square of the tangent length equals the product of the secant's entire length and its external segment.

Step2: Define variables and set up equation

Let $PQ = x$. The entire secant length is $PN = x + 4$, tangent length $MN = 6$.
$$6^2 = 4 \times (x + 4)$$

Step3: Simplify and solve for x

Calculate $6^2 = 36$, expand right side:
$$36 = 4x + 16$$
Subtract 16 from both sides:
$$36 - 16 = 4x$$
$$20 = 4x$$
Divide by 4:
$$x = \frac{20}{4} = 5$$

Answer:

5 units