Question

The line segments connecting the points (x1, y1), (x2, y1), and (x2, y2) form a triangle. Is the triangle a right triangle?

Answer 1

Indeed, the given triangle is a right triangle.

Explanation:

A triangle is formed by the following three points:
`A(x,y) = (x_(1), y_(1))`,
`B(x,y) = (x_(2), y_(1))` and
`C(x,y) = (x_(2), y_(2))`. Then, we construct the following vectors:

`\overrightarrow {BA} = (x_(2)-x_(1), y_(1)-y_(1))`

`\overrightarrow{BA} = (x_(2)-x_(1), 0)` (1)

`\overrightarrow{BC} = (x_(2)-x_(2), y_(2) - y_(1))`

`\overrightarrow{BC} = (0, y_(2)-y_(1))` (2)

If triangle ABC is a right triangle, then
`\overrightarrow{BA}\,\bullet\,\overrightarrow{BC} = 0`. By (1) and (2) we have this expression:

`(x_(2) - x_(1))\cdot (0) + (0)\cdot (y_(2) - y_(1)) = 0`

Therefore, the given triangle is a right triangle.