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Properties of Triangle
Properties of Triangle
Medium

Question

Let the area of a △PQR\triangle P Q R with vertices P(5,4),Q(−2,4)P(5,4), Q(-2,4) and R(a,b)R(a, b) be 35 square units. If its orthocenter and centroid are O(2,145)O\left(2, \frac{14}{5}\right) and C(c,d)C(c, d) respectively, then c+2dc+2 d is equal to

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Solution

 Equation of lines QR=5x+2y+2=0 Equation of lines PR=10x−3y−38=0∴ Point R(2,−6) Centroid =(5−2+23,4+4−63)=(53,23)c+2d=53+43=3\begin{aligned} & \text { Equation of lines } Q R=5 x+2 y+2=0 \\ & \text { Equation of lines } P R=10 x-3 y-38=0 \\ & \therefore \text { Point } R(2,-6) \\ & \text { Centroid }=\left(\frac{5-2+2}{3}, \frac{4+4-6}{3}\right) \\ & =\left(\frac{5}{3}, \frac{2}{3}\right) \\ & c+2 d=\frac{5}{3}+\frac{4}{3}=3 \end{aligned}

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