Straight Lines
Equilateral triangle with given constraints
nta_pyq_2023_jan
Grade 11
Question:
Let B and C be the two points on the line $y + x = 0$ such that B and C are symmetric with respect to the origin. Suppose A is a point on $y - 2x = 2$ such that $\triangle ABC$ is an equilateral triangle. Then, the area of the $\triangle ABC$ is
$3\sqrt{3}$
$2\sqrt{3}$
$\dfrac{8}{\sqrt{3}}$
$\dfrac{10}{\sqrt{3}}$
Step-by-Step Solution
Key Concept: B and C are $(-t, t)$ and $(t, -t)$ for some $t > 0$. A lies on $y - 2x = 2$ and on the perpendicular bisector of BC (which is $y = x$). Find A, then compute height and area.
$A = (-2,-2)$. Height $h = \frac{4}{\sqrt{2}}$. Area $= \frac{\sqrt{3}}{4} \cdot \frac{h^2}{\sin^2 60°} = \frac{8}{\sqrt{3}}$.
Correct Answer: 3