Question

In fig
Ray YM is the bisector of $\angle \mathrm{XYZ}$ where $\mathrm{XY}\cong \mathrm{YZ}$
Find the relation between XM and MZ

Answer 1

Given: Ray YM is the bisector of $\angle \mathrm{XYZ}$ where $\mathrm{XY}\cong \mathrm{YZ}$.

To find: Relation between XM and MZ

We know that in a triangle, the angle bisector divides the side opposite to the angle in the ratio of the remaining sides.

$$\begin{gathered} \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \frac{\mathrm{XM}}{\mathrm{MZ}}=\frac{\mathrm{XY}}{\mathrm{YZ}} \\ \mathrm{Also}, \\ \mathrm{XY}\cong \mathrm{YZ} \\ \Rightarrow \frac{\mathrm{XM}}{\mathrm{MZ}}=1 \\ \Rightarrow \mathrm{seg}\mathrm{XM}\cong \mathrm{seg}\mathrm{MZ} \end{gathered}$$