Explanation:
To prove that the medians of an equilateral triangle are equidistant, we need to show that the medians intersect at a point that is equidistant from all three vertices of the triangle.
Let's first find the midpoints of the sides of the triangle:
Midpoint of the side joining (1, -1) and (-1, 1):
Midpoint = ((1 + (-1)) / 2, (-1 + 1) / 2) = (0, 0)
Midpoint of the side joining (1, -1) and (√3, √3):
Midpoint = ((1 + √3) / 2, (-1 + √3) / 2)
Midpoint of the side joining (-1, 1) and (√3, √3):
Midpoint = ((-1 + √3) / 2, (1 + √3) / 2)
Now, let's find the equation of the medians passing through these midpoints:
Median passing through (0, 0):
The equation of the median passing through (0, 0) and (√3, √3) can be found by using the midpoint formula:
Midpoint = ((0 + √3) / 2, (0 + √3) / 2) = (√3/2, √3/2)
The equation of the median is y = x (since it passes through the origin and the midpoint (√3/2, √3/2)).
Median passing through ((1 + √3) / 2, (-1 + √3) / 2):
The equation of the median passing through ((1 + √3) / 2, (-1 + √3) / 2) and (-1, 1) can be found using the midpoint formula:
Midpoint = (((1 + √3) / 2 - 1) / 2, ((-1 + √3) / 2 + 1) / 2) = ((√3 - 1) / 2, (√3 + 1) / 2)
The equation of the median is y = -x + √3 (since it passes through the midpoint ((√3 - 1) / 2, (√3 + 1) / 2) and (-1, 1)).
Now, we can find the point of intersection of these two medians:
y = x (Equation 1)
y = -x + √3 (Equation 2)
Equating Equation 1 and Equation 2:
x = -x + √3
2x = √3
x = √3/2
Substitute the value of x into Equation 1:
y = √3/2
So, the point of intersection of the medians is (√3/2, √3/2), which is equidistant from all three vertices of the equilateral triangle.
Therefore, the medians of an equilateral triangle are equidistant.