Question

Triangle ABC has vertices A(0,4), B(1,2), and C(4,6). Determine whether ABC is a right triangle. Explain

Answer 1

For it to be a right triangle, it must have a right angle.
Right angles are formed by perpendicuar lines.
If we find that two of these lines are perpendicular, we will know it is a rt. triangle.

Perpendicular lines have slopes which are opposite reciprocals (like 3 and -1/3)

To find the slope between two points, use the formula $m=\frac{rise}{run}=\frac{y_2-y_1}{x_2-x_1}$.

For line AB:
$m=\frac{2-4}{1-0}=\frac{-2}{1}=-2$

For line AC:
$m=\frac{6-4}{4-0}=\frac{2}{4}=\frac{1}2$

For line BC:
$m=\frac{6-2}{4-1}=\frac{4}{3}$

-2 and 1/2 are opposite reciprocals, so this must be a right triangle.

Another way to do this would be to use the distance formula to find the three side lengths and see if the Pythagorean Theorem applies.