Coordinate Geometry
Angle bisector feet; slope of lines making 45° angle
MMTS_Full_Test_09
Grade 12
Question:
In $\triangle ABC$, $X$ and $Y$ are the feet of perpendiculars drawn from $A$ to the internal angle bisectors of $B$ and $C$. The slopes of lines making $45°$ with line $XY$ are (Given slope of $BC = 2$)
(A) $2,\ -\frac{1}{2}$
(B) $-3,\ \frac{1}{3}$
(C) $4,\ -\frac{1}{4}$
(D) $5,\ -\frac{1}{5}$
Step-by-Step Solution
Key Concept: Show that $XY \parallel BC$ (the feet of perpendiculars from $A$ to the angle bisectors of $B$ and $C$ lie on a line parallel to $BC$). Then find slopes of lines making $45°$ with this line.
$XY \parallel BC$, slope $=2$. Lines at $45°$: $m=-3$ or $m=\frac{1}{3}$.
Correct Answer: (B) $-3,\ \frac{1}{3}$