Problem
GEO-B1-M03-P009 Prove the Midline Theorem
#9
★★☆☆☆ Level 2 of 5
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AB\) and \(AC\). Prove that \(MN\parallel BC\) and \(MN=\frac{1}{2}BC\).
Prove that \(\triangle AMN\) and \(\triangle ABC\) are similar.
Since \(M\) and \(N\) are midpoints, \(\frac{AM}{AB}=\frac{AN}{AC}=\frac{1}{2}\). Angle \(A\) is common. By SAS similarity, triangles \(AMN\) and \(ABC\) are similar with ratio \(\frac{1}{2}\). Therefore \(MN\) corresponds to \(BC\), so \(MN\parallel BC\) and \(MN=\frac{1}{2}BC\).
Proving the midline theorem through similarity prepares later median problems.