Properties of TriangleMedium
Question
In an equilateral triangle PQR , let the vertex P be at $(3,5)$ and the side QR be along the line $x + y = 4$. If the orthocentre of the triangle $PQR$ is $(\alpha,\beta)$, then $9(\alpha + \beta)$ is equal to :
Options
A.16
B.27
C.36
D.48
Solution
Sol. [Key Concept: For an equilateral triangle the orthocentre is the centroid; find the foot on $QR$ (line $x + y = 4$) and average the vertices.]
$$Q:\frac{x - 3}{1} = \frac{y - 5}{1} = - \frac{(3 + 5 - 4)}{1 + 1} $$Q(1, 3)
So orthocenter is $\left( \frac{2 + 3}{3},\frac{6 + 5}{3} \right)$
$$\left( \frac{5}{3},\frac{11}{3} \right) $$So $9(\alpha + \beta) = 3(16) = 48$
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