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Question 191409: Two vertices of a triangle in Quadrant 1 are (2,0) and (13,0).The y-coordinate of the third vertex is 2. What is the x-coordinate if the triangle is a right triangle?
A.2 B.13 C.2 or 13 D.11 or 2
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The two given vertices are on the x-axis. We know that because the y-coordinate of both points is 0. That means one side of the triangle is a horizontal line. So we either have the right angle vertex at either end of the horizontal line, or the right angle at the unspecified vertex.
If the right angle is at (2,0), then the adjacent side must be a vertical line and the x-coordinate of ANY point on that vertical line must then be 2. So the point (2,2) is a possibility.
A similar argument shows that the point (2,13) is also a possibility. So now we know that answer A, while correct, is definitely NOT the whole story. Likewise answer B. Also, since answer D does not include 13, it doesn't tell the whole story either. Hence, the best answer so far is C.
I won't go into it here, but multiple applications of Pythagoras' Theorem can prove conclusively that the point (11, 2) is NOT a right angled vertex with the line segment (2,0) to (13,0) being the hypotenuse. However, there is a point somewhere in the first Quadrant that will form a right triangle with the given segment as the hypotenuse, it is just that the coordinates of that point are not among your selections.
John
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