Question

find an and nc in the diagram below
an = type your answer...
nc = type your answer...

Explanation:

Step1: Use the mid - segment theorem

Since \(MN\) is parallel to \(BC\) and \(M\) is the mid - point of \(AB\) (because \(AM = 6\) and \(MB=18 - 6=12\) is not correct, actually by the mid - segment theorem in a triangle, if a line segment connects the mid - points of two sides of a triangle, then it is parallel to the third side and half its length. Here, since \(MN\parallel BC\), we have \(\frac{AN}{NC}=\frac{AM}{MB}\). But also, if \(MN\) is a mid - segment (assuming \(M\) is the mid - point of \(AB\) as \(AM = 6\) and \(AB=18\), so \(M\) is the mid - point), then \(AN = NC\) (by the converse of the mid - segment theorem). So \(7x−1=2x + 1\)

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(7x-2x-1=2x - 2x+1\), so \(5x-1 = 1\). Add \(1\) to both sides: \(5x-1 + 1=1 + 1\), then \(5x=2\), and \(x=\frac{2}{5}\)

Step3: Find \(AN\) and \(NC\)

Substitute \(x = \frac{2}{5}\) into \(AN=2x + 1\): \(AN=2\times\frac{2}{5}+1=\frac{4}{5}+1=\frac{4 + 5}{5}=\frac{9}{5}=1.8\)
Substitute \(x=\frac{2}{5}\) into \(NC = 7x-1\): \(NC=7\times\frac{2}{5}-1=\frac{14}{5}-1=\frac{14 - 5}{5}=\frac{9}{5}=1.8\)

Answer:

\(AN = 1.8\), \(NC = 1.8\)