Straight Lines
Triangle — Angle Conditions and Line Intersection
nta_pyq_2024_apr
Grade 11
Question:
Consider a triangle $ABC$ having the vertices $A(1,2)$, $B(\alpha,\beta)$ and $C(\gamma,\delta)$ and angles $\angle ABC=\dfrac{\pi}{6}$ and $\angle BAC=\dfrac{2\pi}{3}$. If the points $B$ and $C$ lie on the line $y=x+4$, then $\alpha^2+\gamma^2$ is equal to _____
Step-by-Step Solution
Key Concept: Line $BC$: $y=x+4$, slope $m=1$. At $A(1,2)$: $\angle BAC=2\pi/3$. The two lines $AB$ and $AC$ make angle $\pi/6$ with $y=x+4$. Use the angle formula to find slopes of $AB$ and $AC$, then intersect with $y=x+4$ to get $B$ and $C$.
$\alpha^2+\gamma^2=14$.
Correct Answer: 14