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 an external point \(N\) 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 \(NP=NQ + PQ=4 + x\), \(NM = 6\) and \(NQ = 4\).

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, maybe I made a mistake. Wait, the secant segment is \(NP\) and the external part is \(NQ\). Wait, the correct formula is \(NM^{2}=NQ\times NP\), where \(NP\) is the entire secant length (from \(N\) through \(Q\) to \(P\)) and \(NQ\) is the external part. Wait, but if \(PQ=x\), then \(NP=NQ + PQ=4 + x\). But let's check again. Wait, maybe the formula is \(NM^{2}=NQ\times( NQ + PQ)\). So \(6^{2}=4\times(4 + x)\) => \(36 = 16+4x\) => \(4x=36 - 16=20\) => \(x = 5\)? But wait, the options are 4,5,6,9. Wait, but maybe I misread the diagram. Wait, maybe \(PQ\) is the secant? Wait, no, the tangent is \(NM = 6\), the external segment \(NQ = 4\), and the secant is \(NP\), so \(PQ=NP - NQ\). Wait, let's re - derive the tangent - secant theorem. The tangent - secant theorem: If a tangent from \(N\) to the circle touches at \(M\) and a secant from \(N\) passes through the circle, intersecting it at \(Q\) (closer to \(N\)) and \(P\) (farther from \(N\)), then \(NM^{2}=NQ\times NP\). Let \(PQ = x\), so \(NP=NQ + PQ=4 + x\). Then \(6^{2}=4\times(4 + x)\) => \(36=16 + 4x\) => \(4x = 20\) => \(x = 5\)? But wait, the answer options have 5 as an option. Wait, but let's check again. Wait, maybe the diagram is such that \(NQ = 4\), \(NM = 6\), and we need to find \(PQ\). Wait, maybe I made a mistake in the formula. Wait, the correct formula is \(NM^{2}=NQ\times NP\), where \(NP\) is the length from \(N\) to \(P\), and \(NQ\) is from \(N\) to \(Q\). So \(NP=NQ + PQ\). So solving \(36 = 4\times(4 + PQ)\) gives \(PQ = 5\). But wait, the option is 5 units. Wait, but let's check again. Wait, maybe the formula is \(NM^{2}=NQ\times( NQ+PQ)\), so \(36 = 4\times(4 + PQ)\) => \(4 + PQ=9\) => \(PQ = 5\). Yes, that's correct.

Answer:

5 units