Do the points $A (3, 2), B (- 2, - 3)$ and $C (2, 3)$ form a triangle? If so, name the type of triangle formed.

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Solution

We have, using distance formula $\sqrt{(x_{2} - x_{1}) + (y_{2} - y_{1})}$
$A B = \sqrt{{(- 2 - 3)}^{2} + {(- 3 - 2)}^{2}} = \sqrt{25 + 25} = \sqrt{50}$
$B C = \sqrt{{(2 + 2)}^{2} + {(3 + 3)}^{2}} = \sqrt{16 + 36} = \sqrt{52}$
and $A C = \sqrt{{(2 - 3)}^{2} + {(3 - 2)}^{2}} = \sqrt{1 + 1} = \sqrt{2}$
$C l e a r l y, A B + B C > A C, A C + B C > A B a n d A B + A C > B C .$
Therefore, points A,B and C form a triangle.
$A l s o, {B C}^{2} = {A B}^{2} + {A C}^{2}$
Therefore, $△ A B C$ is a right triangle, right angled at A.

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Do the points A(3,2),B(−2,−3) and C(2,3) form a triangle? If so, name the type of triangle formed.

Do the points $A (3, 2), B (- 2, - 3)$ and $C (2, 3)$ form a triangle? If so, name the type of triangle formed.