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Properties of Triangle
Properties of Triangle
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Question

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is :

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Solution

In a triangle, orthocentre, centroid and circumcenter are collinear and centroid divides orthocenter and circumcenter in 2 : 1 ratio. Here AB = 2 BC AB=(3+3)2+(35)2AB = \sqrt {{{\left( {3 + 3} \right)}^2} + {{\left( {3 - 5} \right)}^2}} =36+4= \sqrt {36 + 4} =40= \sqrt {40} =210= 2\sqrt {10} \therefore\,\,\,\, BC =10= \sqrt {10} \therefore\,\,\,\, AB + BC =210+10= 2\sqrt {10} + \sqrt {10} =310= 3\sqrt {10} AC is the diameter of the circle. \therefore\,\,\, Radius =12 = {1 \over 2}\,\, AC =12×310= {1 \over 2} \times 3\sqrt {10} =352= 3\sqrt {{5 \over 2}}

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