if a(5 -10) b(-3 -2)c(-1 8) are the vertices of triangle abc find the length of the median through a and the coordinates of the centroid
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Let A (x\(_{1}\), y\(_{1}\)), B (x\(_{2}\), y\(_{2}\)) and C (x\(_{3}\), y\(_{3}\)) are the three vertices of the ∆ABC .
Let D be the midpoint of side BC.
Since, the coordinates of B (x\(_{2}\), y\(_{2}\)) and C (x\(_{3}\), y\(_{3}\)), the coordinate of the point D are (\(\frac{x_{2} + x_{3}}{2}\), \(\frac{y_{2} + y_{3}}{2}\)).
Let G(x, y) be the centroid of the triangle ABC.
Then, from the geometry, G is on the median AD and it divides AD in the ratio 2 : 1, that is AG : GD = 2 : 1.
Therefore, x = \(\left \{\frac{2\cdot \frac{(x_{2} + x_{3})}{2} + 1 \cdot x_{1}}{2 + 1}\right \}\) = \(\frac{x_{1} + x _{2} + x_{3}}{3}\)
y = \(\left \{\frac{2\cdot \frac{(y_{2} + y_{3})}{2} + 1 \cdot y_{1}}{2 + 1}\right \}\) = \(\frac{y_{1} + y _{2} + y_{3}}{3}\)
Therefore, the coordinate of the G are (\(\frac{x_{1} + x _{2} + x_{3}}{3}\), \(\frac{y_{1} + y _{2} + y_{3}}{3}\))
Hence, the centroid of a triangle whose vertices are (x\(_{1}\), y\(_{1}\)), (x\(_{2}\), y\(_{2}\)) and (x\(_{3}\), y\(_{3}\)) has the coordinates (\(\frac{x_{1} + x _{2} + x_{3}}{3}\), \(\frac{y_{1} + y _{2} + y_{3}}{3}\)).
Note: The centroid of a triangle divides each median in the ratio 2 : 1 (vertex to base).
Solved examples to find the centroid of a triangle:
1. Find the co-ordinates of the point of intersection of the medians of trangle ABC; given A = (-2, 3), B = (6, 7) and C = (4, 1).
Solution:
Here, (x\(_{1}\) = -2, y\(_{1}\) = 3), (x\(_{2}\) = 6, y\(_{2}\) = 7) and (x\(_{3}\) = 4, y\(_{3}\) = 1),
Let G (x, y) be the centroid of the triangle ABC. Then,
x = \(\frac{x_{1} + x _{2} + x_{3}}{3}\) = \(\frac{(-2) + 6 + 4}{3}\) = \(\frac{8}{3}\)
y = \(\frac{y_{1} + y _{2} + y_{3}}{3}\) = \(\frac{3 + 7 + 1}{3}\) = \(\frac{11}{3}\)
Therefore, the coordinates of the centroid G of the triangle ABC are (\(\frac{8}{3}\), \(\frac{11}{3}\))
Thus, the coordinates of the point of intersection of the medians of triangle are (\(\frac{8}{3}\), \(\frac{11}{3}\)).
2. The three vertices of the triangle ABC are (1, -4), (-2, 2) and (4, 5) respectively. Find the centroid and the length of the median through the vertex A
Let D be the midpoint of side BC.
Since, the coordinates of B (x\(_{2}\), y\(_{2}\)) and C (x\(_{3}\), y\(_{3}\)), the coordinate of the point D are (\(\frac{x_{2} + x_{3}}{2}\), \(\frac{y_{2} + y_{3}}{2}\)).
Let G(x, y) be the centroid of the triangle ABC.
Then, from the geometry, G is on the median AD and it divides AD in the ratio 2 : 1, that is AG : GD = 2 : 1.
Therefore, x = \(\left \{\frac{2\cdot \frac{(x_{2} + x_{3})}{2} + 1 \cdot x_{1}}{2 + 1}\right \}\) = \(\frac{x_{1} + x _{2} + x_{3}}{3}\)
y = \(\left \{\frac{2\cdot \frac{(y_{2} + y_{3})}{2} + 1 \cdot y_{1}}{2 + 1}\right \}\) = \(\frac{y_{1} + y _{2} + y_{3}}{3}\)
Therefore, the coordinate of the G are (\(\frac{x_{1} + x _{2} + x_{3}}{3}\), \(\frac{y_{1} + y _{2} + y_{3}}{3}\))
Hence, the centroid of a triangle whose vertices are (x\(_{1}\), y\(_{1}\)), (x\(_{2}\), y\(_{2}\)) and (x\(_{3}\), y\(_{3}\)) has the coordinates (\(\frac{x_{1} + x _{2} + x_{3}}{3}\), \(\frac{y_{1} + y _{2} + y_{3}}{3}\)).
Note: The centroid of a triangle divides each median in the ratio 2 : 1 (vertex to base).
Solved examples to find the centroid of a triangle:
1. Find the co-ordinates of the point of intersection of the medians of trangle ABC; given A = (-2, 3), B = (6, 7) and C = (4, 1).
Solution:
Here, (x\(_{1}\) = -2, y\(_{1}\) = 3), (x\(_{2}\) = 6, y\(_{2}\) = 7) and (x\(_{3}\) = 4, y\(_{3}\) = 1),
Let G (x, y) be the centroid of the triangle ABC. Then,
x = \(\frac{x_{1} + x _{2} + x_{3}}{3}\) = \(\frac{(-2) + 6 + 4}{3}\) = \(\frac{8}{3}\)
y = \(\frac{y_{1} + y _{2} + y_{3}}{3}\) = \(\frac{3 + 7 + 1}{3}\) = \(\frac{11}{3}\)
Therefore, the coordinates of the centroid G of the triangle ABC are (\(\frac{8}{3}\), \(\frac{11}{3}\))
Thus, the coordinates of the point of intersection of the medians of triangle are (\(\frac{8}{3}\), \(\frac{11}{3}\)).
2. The three vertices of the triangle ABC are (1, -4), (-2, 2) and (4, 5) respectively. Find the centroid and the length of the median through the vertex A
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