Straight Line and Pair of Straight Lines 2 Question 4
4. 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
Statement II In any triangle, bisector of an angle divides the triangle into two similar triangles. (2007, 3M)
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Answer:
Correct Answer: 4. (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 $\angle P O Q$.
$$ \begin{array}{lll} \therefore & & \frac{P R}{R Q}=\frac{O P}{O Q} \\ \Rightarrow & \frac{P R}{R Q}=\frac{\sqrt{2^{2}+2^{2}}}{\sqrt{1^{2}+2^{2}}}=\frac{2 \sqrt{2}}{\sqrt{5}} \end{array} $$
