Question
Given below are some triangles and lengths of line segments. Identify in which figures, ray PM is the bisector of ∠OPR.
✓
Answer
Theorem: The bisector of an angle of a triangle divides the side opposite to the angle in the ratio of the remaining sides.Therefore, we’ll find the ratio for all the triangle.Hence, for
$\text { (1) } \frac{Q M}{M R}=\frac{3.5}{1.5} $
$=2.33$
And $\frac{ QP }{ PR }=\frac{7}{3}$
$=2.33 $
$\Rightarrow \frac{ QM }{ MR }=\frac{ QP }{ PR }$
$\Rightarrow \ln (1)$, ray PM is a bisector.
$\text { (2) } \frac{ RM }{ MQ }=\frac{6}{8} $
$=0.75$
And $\frac{ RP }{ PQ }=\frac{7}{10}$
$=0.7 $
$\Rightarrow \frac{ RM }{ MQ } \neq \frac{ RP }{ PQ }$
$\Rightarrow \ln (2)$, ray PM is not a bisector.
$\text { (3) } \frac{ RM }{ MQ }=\frac{4}{3.6} $
$=1.1$
And $\frac{R P}{P Q}=\frac{10}{9}$
$=1.11$
$\Rightarrow \frac{ RM }{ MQ }=\frac{ RP }{ PQ }$
$\Rightarrow \ln (3)$, ray PM is a bisector.
$\text { (1) } \frac{Q M}{M R}=\frac{3.5}{1.5} $
$=2.33$
And $\frac{ QP }{ PR }=\frac{7}{3}$
$=2.33 $
$\Rightarrow \frac{ QM }{ MR }=\frac{ QP }{ PR }$
$\Rightarrow \ln (1)$, ray PM is a bisector.
$\text { (2) } \frac{ RM }{ MQ }=\frac{6}{8} $
$=0.75$
And $\frac{ RP }{ PQ }=\frac{7}{10}$
$=0.7 $
$\Rightarrow \frac{ RM }{ MQ } \neq \frac{ RP }{ PQ }$
$\Rightarrow \ln (2)$, ray PM is not a bisector.
$\text { (3) } \frac{ RM }{ MQ }=\frac{4}{3.6} $
$=1.1$
And $\frac{R P}{P Q}=\frac{10}{9}$
$=1.11$
$\Rightarrow \frac{ RM }{ MQ }=\frac{ RP }{ PQ }$
$\Rightarrow \ln (3)$, ray PM is a bisector.
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