Find the coordinates of the Sercombe center of a triangle with the given vertices L(3,-6) M(5,-3) N(8,-6)
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As a result, the Sercombe centre of the triangle is (3, -1).
As per the question given,
To find the coordinates of the Sercombe center of the triangle with vertices L(3,-6), M(5,-3), and N(8,-6), we need to find the intersection point of the three medians of the triangle.
The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. We first find the midpoints of the sides LM, MN, and NL:
Midpoint of LM: ((3+5)/2, (-6-3)/2) = (4, -4.5)
Midpoint of MN: ((5+8)/2, (-3-6)/2) = (6.5, -4.5)
Midpoint of NL: ((8+3)/2, (-6-6)/2) = (5.5, -6)
Next, we find the equations of the medians that pass through these midpoints. Since the median through L passes through the midpoint of MN, we can find its equation as follows:
Slope of MN: (-3-(-6))/(5-8) = 1/3
Midpoint of MN: (6.5, -4.5)
Equation of median through L: y - (-6) = (1/3)(x - 3)
Simplifying this equation, we get: y = (1/3)x - 5
Similarly, we can find the equations of the other two medians:
Equation of median through M: y - (-3) = (-3/2)(x - 5)
Simplifying this equation, we get: y = (-3/2)x + 13.5
Equation of median through N: y - (-6) = (1/2)(x - 8)
Simplifying this equation, we get: y = (1/2)x - 2
Now we need to find the point of intersection of these three medians. We can solve the system of equations formed by these three median equations to find this point.
Solving, we get: (3, -1)
Therefore, the Sercombe centre of the triangle is (3, -1).
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