Let the area of a $ riangle PQR$ with vertices $P(5,4)$, $Q(-2,4)$ and $R(a,b)$ be 35 square units. If its orthocenter and centroid are $O\left(2,\frac{14}5 ight)$ and $C(c,d)$ respectively, then $c+2d$ is equal to

Question

Let the area of a $	riangle PQR$ with vertices $P(5,4)$, $Q(-2,4)$ and $R(a,b)$ be 35 square units. If its orthocenter and centroid are $O\left(2,\frac{14}5ight)$ and $C(c,d)$ respectively, then $c+2d$ is equal to

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] Area $=\frac12\cdot7\cdot|b-4|=35\Rightarrow|b-4|=10$. $PQ$ is horizontal, so the altitude from $R$ is the line $x=a$ and the orthocentre has $x=2$, so $a=2$. The orthocentre's $y$-coordinate is $4-\frac{(5-2)(-2-2)}{b-4}=4+\frac{12}{b-4}=\frac{14}5\Rightarrow b-4=-10$, $b=-6$. The centroid is $\left(\frac{5-2+2}3,\frac{4+4-6}3\right)=\left(\frac53,\frac23\right)$, so $c+2d=\frac53+\frac43=3$. Answer (4).

Q. Let the area of a $\triangle PQR$ with vertices $P(5,4)$, $Q(-2,4)$ and $R(a,b)$ be 35 square units. If its orthocenter and centroid are $O\left(2,\frac{14}5\right)$ and $C(c,d)$ respectively, then $c+2d$ is equal to

Explanation

Reviewed by Dr. Abubacker

Want to attempt the full paper?