If a vertex, the , circumcenter, and the centroid of a triangle are ( 0, 0 ), (3, 4) and (6, 8) respectively, then the triangle must be
a. a right angles triangle 
b. an equilateral triangle
c. an isosceles triangle
d. a right-angled isosceles triangle

Sol: Clearly, ( 0, 0 ), (3, 4) and (6, 8) are collinear. So, the circumcenter M and the centroid G are on the median which is also the perpendicular bisector of the side. So the triangle must be isosceles. 

How did we come to know they're on the median and also the perpendicular bisector? (I couldn't understand it through the triangle I plotted. Please make a figure if that makes things easier)


Dear Student,
Please find below the solution to the asked query:

Consider isosceles triangle ABC, where AB=AC and AD is median.In ACD and ABDAC=ABC=CCD=BD As ADis medianHence ACDABDADC=ADBAlso ADC+ADB=180° Linear Pair AxiomHence ADC=ADB=90°Hence ADis perpendicular bisector of BC.Hence centroid, circumcentre and vertex A will lie on AD.Hence isosceles triangle.

Hope this information will clear your doubts about this topic.

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