Straight Lines
Angle Bisector — Finding α+2β
nta_pyq_2024_jan
Grade 11
Question:
In a $\triangle ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x-y=2$. If $2AB=BC$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha,\beta)$, then $\alpha+2\beta$ is equal to
Step-by-Step Solution
Key Concept: Reflect $A=(4,6)$ over the bisector $y=x$ to get $A'=(6,4)$. The line $BA'$ intersects $AC$ at $C$. Use $2AB=BC$ to find $B$. Since $B$ lies on $y=x$, $B=(\beta,\beta)$... actually $B=(\alpha,\beta)$ with $\beta=\alpha$. Use section formula with $AD:DC=1:2$.
$\alpha=14,\beta=14$. $\alpha+2\beta=42$.
Correct Answer: 1