Question 1195474
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Determine if the points (3,5), (3,3) and (2,4) are the vertices of a right triangle. 
Explain your rationale.
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Points A = (3,5) and B = (3,3) have the same x-coordinate.


It means that both points A and B lie on vertical line x= 3, parallel to y-axis.



Next,      points  A = (3,5) and C = (2,4)  have different y-coordinates; hence, line AC is not horizontal.

Similarly, points  B = (3,3) and C = (2,4)  have different y-coordinates; hence, line BC is not horizontal.



Thus, line AB is vertical, but angles A and B are not right angles.


So, we only should check if lines AC and BC are perpendicular.



Line AC has the slope  {{{(4-5)/(2-3)}}} = {{{(-1)/(-1)}}} = 1.

Line BC has the slope  {{{(4-3)/(2-3)}}} = {{{1/(-1)}}} = -1.



Thus lines AC and BC are perpendicular, since their slopes are 1 and -1, i.e. negatively reciprocal 

     (their product is  1 * (-1) = -1).


<U>ANSWER</U>.  Angle C is the right angle.
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Solved.