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Two vertices of a triangle are (0, -6) and (4, 0), and the medians intersect at (0, -2). Find the third vertex of a triangle.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
The first thing that we need to do is to get the coordinates of the midpoint of line segment AB which is D as follows
Hence, the coordinates of the midpoint of line segment AB is D (2, -3).
If OD is the distance of the point of intersection of the medians to the midpoint of the opposite side which is r, then OC is the distance of the point of intersection of the medians to the opposite vertex which is 2r.
The third vertex of a triangle can be solved by using division of line segment formula as follows
where k is the ratio of two line segments. If the value of k is a fraction, then it is a division of line segment and if the value of k is a whole number greater than one, then it is an extension of line segment. In this problem, we will do the extension of a line segment since we are solving for the third vertex of a triangle.
The value of k is 1 (from OD) plus 2 (from OC) equals 3. Substitute the values of the coordinates of points O and D to the above equation, we have
Therefore, the third vertex of a triangle is C (-4, 0).