If a vertex, the , circumcenter, and the centroid of a triangle are ( 0, 0 ), (3, 4) and - Maths - Straight Lines

Question

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)

Shivam Mishra · Accepted Answer

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

<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/ck-files/ck_5813a90c18d65%20jpg"
style="width: 349px; height: 339px;">
Consider isosceles triangle ABC, where AB=AC and AD is median
In △ACD and △ABDAC=AB∠C=∠CCD=BD As AD is medianHence △ACD≅△ABD⇒∠ADC=∠ADBAlso ∠ADC+∠ADB=180° Linear Pair AxiomHence ∠ADC=∠ADB=90°Hence AD is perpendicular bisector of BC
Hence centroid, circumcentre and vertex A will lie on AD
Hence isosceles triangle

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