Alg6.17

Show that the triangle formed by the points(4,4),(3,5) and (-1,-1) is a right angled traingle.

Let the given points be A,B,C are the vertices of the given triangle.

Now finding the distance between the vertices or the length of the sides,

${\bar A}B={\sqrt {(4-3)^{2}+(4-5)^{2}}},{\bar B}C={\sqrt {(3+1)^{2}+(5+1)^{2}}},{\bar C}A={\sqrt {(-1-4)^{2}+(-1-4)^{2}}}\,$

${\bar A}B={\sqrt {2}},{\bar B}C={\sqrt {52}},{\bar C}A={\sqrt {50}}\,$

The realation among the three sides is like

$({\bar B}C)^{2}=({\bar A}B)^{2}+({\bar C}A)^{2}\,$

This is the essential condition of a right angled triangle.

Hence the given points form the vertices of a right angled traingle.