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prove that the medians of an equilateral triangle having vertices (1,-1) , (-1,1) and (√3,√3) are equidistant.​

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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.

User John Kloian
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