Answer:
(a) (2, 2) — right triangle
(b) (0, 4) — not
(d) (-2, 1) — right triangle
Explanation:
You want to know if any of the points C(2, 2), C(0, 4), or C(-2, 1) together with A(-2, 4) and B(-1, 1) will form a right triangle.
Math
There are several ways one can check to see if three points form a right triangle. Perhaps one of the easiest is to look at the vectors between pairs of points. For example, vector AB = B -A = (-1, 1) -(-2, 4) = (-1+2, 1-4) = (1, -3).
Then, we can determine if two vectors are perpendicular a couple of ways. One way is to form the ratios y/x for two vectors, and check to see if any pair of ratios has a product of -1. This can be inconclusive if one of the vectors has a zero component.
Another way to check for perpendicularity is to form the "dot product" of two vectors. For vectors (a, b) and (c, d), that is the sum (ac+bd). If that value is zero, the two vectors are perpendicular.
Spreadsheet
Algebraically, we can check to see if a right triangle is defined by ...
- forming vectors from pairs of points
- finding the dot product of pairs of vectors.
This math is accomplished in the spreadsheet in the second attachment. The dot products that are zero are highlighted in bright green.
Points C(2, 2) and C(-2, 1) will form right triangles with points A and B.
Graph
Perhaps the easiest way to determine if a right triangle is formed from these points is to graph them. The first attachment shows the triangles.
Points C(2, 2) and C(-2, 1) will form right triangles with points A and B.
If the point coordinates are farther apart, it may be more difficult to tell if a right angle is formed. For these points, this method works well.
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