Median of a triangle
a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side
In geometry, a median of a triangle refers to a line segment drawn from a vertex of a triangle to the midpoint of the opposite side. Each triangle has three medians that intersect at the centroid of the triangle, which is the center of gravity of the triangle.
The median of a triangle is half the length of the side opposite the vertex it originates from. This means that if we know the length of any two sides of the triangle and the length of one median, then we can find the length of the third side and the other two medians using the properties of medians.
To find the length of a median, we can use the following formula:
Median = (1/2) * square root of (2a^2 + 2b^2 – c^2)
where a, b, and c are the lengths of the sides of the triangle and c is the length of the side opposite to the vertex from which the median originates.
Similarly, we can find the length of the third side of the triangle, using the formula:
c = square root of (2a^2 + 2b^2 – 4m^2)
where m is the length of the median.
Knowing the length of all three medians of a triangle (which is the same as knowing the length of any two sides and one median), we can find the area of the triangle using the formula:
Area of triangle = (4/3) * square root of (s(s – m1)(s – m2)(s – m3))
where s is the semiperimeter of the triangle, which is half the sum of the lengths of the sides of the triangle, and m1, m2, and m3 are the lengths of the three medians.
In summary, medians play an important role in geometry and can be used to find the length of sides and area of a triangle.
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