Question

In the figure given above, $AP$ $\perp l$ i.e., $AP$ is the shortest line segment that can be drawn from $A$ to the line $l$. If $PR>PQ$, which of the following is true.

• A. $AQ>AR$
• B. $AR>AQ$
• C. $AQ=2AR$
• D. $AQ=\sqrt{2}AR$

Answer 1

Answer: (B)

$AR>AQ$

Cut off $PS$ $=$ $PQ$ from $PR$ join $AS$
In $\Delta APQ$ and $\Delta APS$
$\angle APQ=\angle APS$ (each ${90}^{\circ }$)
$AP$ $=$ $AP$ .... (common side)
$PQ$ $=$ $PS$
$\therefore \Delta APQ\cong \Delta APS$
$AP$ $=$ $AS$ and $\angle 1=\angle 3$ ....(1)
In $\Delta ARS$, $\angle 3>\angle 2$ ....(2)
[exterior angle property]
from (1) and (2)
$\angle 1>\angle 2$
from $\Delta AQR$, $\angle 1>\angle 2$
$\therefore AR>QR$
$\therefore AR>AQ$