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=(rise)/(run)=(y_2-y_1)/(x_2-x_1)`.

For line AB:

`m=(2-4)/(1-0)=(-2)/(1)=-2`

For line AC:

`m=(6-4)/(4-0)=(2)/(4)=\frac{1}2`

For line BC:

`m=(6-2)/(4-1)=(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.