Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.

Given, the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.
Therefore,
Let A (3,0), B (6,4) and C (-1,3) be the given points.
Now,

$AB space equals space square root of left parenthesis 6 minus 3 right parenthesis squared plus left parenthesis 4 plus 0 right parenthesis squared end root space equals space square root of 3 squared plus 4 squared end root space equals space square root of 9 plus 16 end root space equals square root of 25 BC equals square root of negative left parenthesis 1 minus 6 right parenthesis squared plus left parenthesis 3 minus 4 right parenthesis squared end root space equals square root of left parenthesis negative 7 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared end root space equals space square root of 49 plus 1 end root space equals square root of 50 AC equals square root of left parenthesis negative 1 minus 3 right parenthesis squared plus left parenthesis 3 minus 0 right parenthesis squared end root space equals space square root of left parenthesis negative 4 right parenthesis squared plus 3 squared space end root space equals space square root of 16 plus 9 end root space equals square root of 25 Since space AB equals BC therefore AB squared equals left parenthesis square root of 25 right parenthesis squared space equals 25 BC squared space equals left parenthesis square root of 50 right parenthesis end root squared space equals 50 AC squared space equals space left parenthesis square root of 25 right parenthesis squared space equals 25 therefore space AB squared plus AC squared space equals space BC squared Thus comma space increment ABC space is space right space minus angled space isosceles space triangle$

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