Straight Line And Pair Of Straight Lines Ques 75
- Lines $L _1: y-x=0$ and $L _2: 2 x+y=0$ intersect the line $L _3: y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L _1$ and $L _2$ intersects $L _3$ at $R$.
Statement I The ratio $P R: R Q$ equals $2 \sqrt{2}: \sqrt{5}$.
Because
In any triangle, the bisector of an angle divides the opposite side in the ratio of the adjacent sides, but does not necessarily divide the triangle into two similar triangles. (2007, 3M)
Fill in the Blank
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Answer:
Correct Answer: 75.(c)
Solution:
- It is not necessary that the bisector of an angle will divide the triangle into two similar triangles, therefore, statement II is false.
Now, we verify Statement I.
$\triangle O P Q, O R$ is the internal bisector of $\angle P O Q$.
$$ \begin{array}{lll} \therefore & & \frac{P R}{R Q}=\frac{O P}{O Q} \\ \Rightarrow & \frac{P R}{R Q}=\frac{\sqrt{2^{2}+2^{2}}}{\sqrt{1^{2}+2^{2}}}=\frac{2 \sqrt{2}}{\sqrt{5}} \end{array} $$
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