Lines and intersect the line at and , respectively. The bisector of the acute angle between and intersects at . Statement-1: The ratio equals . Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

Question

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-1: The ratio $P R : R Q$ equals $2 \sqrt{2} : \sqrt{5}$ .

Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles. [2007]

Smart Achievers · Accepted Answer

(3)

$Point of intersection of L_{1} and L_{2} is A (0, 0) .$

$Also P (- 2, - 2), Q (1, - 2)$

$∵$ $A R is the bisector of ∠ P A Q, therefore R divides P Q in the ratio of A P : A Q .$

$i.e., P R : R Q = A P : A Q = 2 \sqrt{2} : \sqrt{5}$

$∴$ $Statement-1 is true.$

$Statement-2 is clearly false.$

Solution

1 Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

2 Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

3 Statement-1 is True, Statement-2 is False

4 Statement-1 is False, Statement-2 is True

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