Question

what is the length of line segment pq?
4 units
5 units
6 units
9 units

Explanation:

Step1: Recall the tangent - secant rule

The tangent - secant rule states that if a tangent from a point \(N\) to a circle touches the circle at \(M\) and a secant from \(N\) passes through the circle intersecting it at \(Q\) and \(P\), then \(NM^{2}=NQ\times NP\). Let \(PQ = x\), then \(NP=NQ + PQ=4 + x\), \(NQ = 4\) and \(NM = 6\).

Step2: Substitute the values into the formula

Substitute \(NM = 6\), \(NQ = 4\) and \(NP=4 + x\) into the formula \(NM^{2}=NQ\times NP\), we get \(6^{2}=4\times(4 + x)\).

Step3: Solve the equation

First, calculate \(6^{2}=36\), so the equation becomes \(36=4\times(4 + x)\). Divide both sides by 4: \(\frac{36}{4}=4 + x\), which simplifies to \(9 = 4+x\). Then subtract 4 from both sides: \(x=9 - 4=5\)? Wait, no, wait. Wait, the question is about \(PQ\)? Wait, no, maybe I made a mistake. Wait, the tangent - secant formula is \(NM^{2}=NQ\times NP\), where \(NP\) is the entire secant length from \(N\) through \(Q\) to \(P\), so \(NP=NQ + PQ\), but maybe the question is about \(PQ\)? Wait, no, the options are 4,5,6,9. Wait, maybe I misread the diagram. Wait, maybe \(NQ = 4\), \(NM = 6\), and we need to find \(PQ\). Wait, let's re - check. The tangent - secant theorem: if a tangent from \(N\) touches the circle at \(M\), and a secant from \(N\) intersects the circle at \(Q\) and \(P\) (with \(Q\) closer to \(N\) than \(P\)), then \(NM^{2}=NQ\times NP\). So \(NM = 6\), \(NQ = 4\), so \(6^{2}=4\times NP\), so \(NP=\frac{36}{4}=9\). Then \(PQ=NP - NQ=9 - 4 = 5\)? But the options have 5 units? Wait, but the question is "What is the length of line segment PQ?". Wait, but let's check again. Wait, maybe the diagram is such that \(NQ = 4\), \(NP=PQ + QN\), so \(NP=PQ + 4\). Then \(6^{2}=4\times(PQ + 4)\), \(36 = 16+4PQ\), \(4PQ=36 - 16 = 20\), \(PQ = 5\). But the options have 5 units as one of the options. Wait, but maybe I made a mistake. Wait, no, the tangent - secant theorem is \(NM^{2}=NQ\times NP\), where \(NP\) is the length from \(N\) to \(P\) (the entire secant), \(NQ\) is from \(N\) to \(Q\) (the external part), and \(NM\) is the tangent. So \(NM = 6\), \(NQ = 4\), so \(NP=\frac{NM^{2}}{NQ}=\frac{36}{4}=9\). Then \(PQ=NP - NQ=9 - 4 = 5\). But the options include 5 units. Wait, but maybe the question is not about \(PQ\) but about something else? Wait, no, the question is "What is the length of line segment PQ?". So if \(NP = 9\) and \(NQ = 4\), then \(PQ=NP - NQ=5\). But wait, the options have 5 units. So the answer should be 5 units? Wait, but let's check the options again. The options are 4 units, 5 units, 6 units, 9 units. So according to the calculation, \(PQ = 5\) units.

Answer:

5 units (B. 5 units)