Question
Write the coordinates of the circumcentre of a triangle whose centroid and orthocentre are at (3, 3) and (-3, 5), respectively.
✓
Answer
We know that the orthocentre, centroid and the circumcentre of a triangle are collinear. Also we know that the distance between orthocentre and centroid is twice the distance between centroid and the circumcentre. Given that centroid of a triangle is G(3, 3), orthocentre is H(-3, 5) Let C(x, y) be the coordinates of the circumcentre of the triangle.
Hence using section formula, we have, $\frac{2\text{x}\times-3\times1}{2+1}=3$ and $\frac{2\times\text{y}+5\times1}{2+1}=3$ $\Rightarrow\frac{2\text{x}-3}{3}=3$ and $\frac{2\text{y}+5}{3}=3$ $\Rightarrow2\text{x}-3=9$ and $2\text{y}+5=9$ $\Rightarrow2\text{x}=9+3$ and $2\text{y}=9-5$ $\Rightarrow2\text{x}=12$ and $2\text{y}=4$ $\Rightarrow\text{x}=6$ and $\text{y}=2$ Thus, the circumc:entre of the triangle is C(6, 2)
Hence using section formula, we have, $\frac{2\text{x}\times-3\times1}{2+1}=3$ and $\frac{2\times\text{y}+5\times1}{2+1}=3$ $\Rightarrow\frac{2\text{x}-3}{3}=3$ and $\frac{2\text{y}+5}{3}=3$ $\Rightarrow2\text{x}-3=9$ and $2\text{y}+5=9$ $\Rightarrow2\text{x}=9+3$ and $2\text{y}=9-5$ $\Rightarrow2\text{x}=12$ and $2\text{y}=4$ $\Rightarrow\text{x}=6$ and $\text{y}=2$ Thus, the circumc:entre of the triangle is C(6, 2)
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