Question

Given below are some triangles and lengths of line segments. Identify in which figures, ray PM is the bisector of ∠QPR.
(1)

(2)

(3)

Answer 1

(1)
$$\begin{gathered} \mathrm{In}△\mathrm{QMP}, \\ \frac{\mathrm{QM}}{\mathrm{QP}}=\frac{3.5}{7} \\ =\frac{1}{2} \end{gathered}$$
$$\begin{gathered} \mathrm{In}△\mathrm{MRP}, \\ \frac{\mathrm{MR}}{\mathrm{RP}}=\frac{1.5}{3} \\ =\frac{1}{2} \end{gathered}$$
$\therefore \frac{\mathrm{QM}}{\mathrm{QP}}=\frac{\mathrm{MR}}{\mathrm{RP}}$
By converse of angle bisector theorem, ray PM is the bisector of ∠QPR.

(2)
$$\begin{gathered} \mathrm{In}△\mathrm{QMP}, \\ \frac{\mathrm{QM}}{\mathrm{QP}}=\frac{8}{10} \\ =\frac{4}{5} \end{gathered}$$
$$\begin{gathered} \mathrm{In}△\mathrm{MRP}, \\ \frac{\mathrm{MR}}{\mathrm{RP}}=\frac{6}{7} \end{gathered}$$
$\therefore \frac{\mathrm{QM}}{\mathrm{QP}}\ne \frac{\mathrm{MR}}{\mathrm{RP}}$
Therefore, ray PM is not the the bisector of ∠QPR.
(3)
$$\begin{gathered} \mathrm{In}△\mathrm{QMP}, \\ \frac{\mathrm{QM}}{\mathrm{QP}}=\frac{3.6}{9} \\ =\frac{2}{5} \end{gathered}$$
$$\begin{gathered} \mathrm{In}△\mathrm{MRP}, \\ \frac{\mathrm{MR}}{\mathrm{RP}}=\frac{4}{10} \\ =\frac{2}{5} \end{gathered}$$
$\therefore \frac{\mathrm{QM}}{\mathrm{QP}}=\frac{\mathrm{MR}}{\mathrm{RP}}$
By converse of angle bisector theorem, ray PM is the bisector of ∠QPR.

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