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In △ABC, the coordinates of vertices A and B are A(−2,4) and B(−1,1).   For each of the given coordinates of vertex C, is  △ABC a right triangle?   Select Right Triangle or Not a Right Triangle for each set of coordinates. Right Triangle Not a Right Triangle C(2,2) Right Triangle – C begin ordered pair 2 comma 2 end ordered pair Not a Right Triangle – C begin ordered pair 2 comma 2 end ordered pair C(0,4) Right Triangle – C begin ordered pair 0 comma 4 end ordered pair Not a Right Triangle – C begin ordered pair 0 comma 4 end ordered pair C(−2,1) Right Triangle – C begin ordered pair negative 2 comma 1 end ordered pair Not a Right Triangle – C begin ordered pair negative 2 comma 1 end ordered pair

User Nichoio
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1 Answer

4 votes

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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In △ABC, the coordinates of vertices A and B are A(−2,4) and B(−1,1).   For each of-example-1
In △ABC, the coordinates of vertices A and B are A(−2,4) and B(−1,1).   For each of-example-2
User Andrei Zhytkevich
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