Question

In triangle ABC, M is mid-point of AB and a straight line through M and parallel to BC cuts AC at N. Find the lenghts of AN and MN, if BC=7cm and AC=5cm.

Answer 1

IN $\Delta AMN$ $\Delta ABC$

Since $MN||BC$

$\angle AMN=\angle ABC$ (Corresponding angles)

$\angle ANM=\angle ACB$ (Corresponding angles)

$\therefore \Delta AMN\sim \Delta ABC$(By $AA similarity criterion)

$\Rightarrow \frac{AM}{AB}=\frac{AN}{AC}=\frac{MN}{BC}\left(CPST\right)$

Since, $M$ is mid-point of $AB$,

$AM=\frac{1}{2}AB,or,\frac{AM}{AB}=\frac{1}{2}or,\frac{AM}{AB}=\frac{AN}{AC}=\frac{1}{2}\frac{AN}{AC}=\frac{1}{2}\frac{AN}{5}=\frac{1}{2}[\because AC=5cm]AN=\frac{5}{2}cm=2.5cm$

Also, $\frac{AM}{AB}=\frac{MN}{BC}=\frac{1}{2}\frac{MN}{7}=\frac{1}{2}[\because BC=7cm]MN=\frac{7}{2}=3.5$

Ans=$AN=2.5cm$ and $MN=3.5cm$