Make Note On Solved Problems On Relation Between Median And Sides Of A Triangle

In the geometrical analysis of a polygon, a triangle, the median of a triangle is a span of line segments joining vertices of the angle to the middle point of the opposite side of the triangle, thereby bisecting the side into two distinct sections. In a triangle, there are three medians in the total which are drawn from each vertex and these medians intersect one another at the centroid. The medians crossing the centroid subdivide the triangle into six smaller triangles. 

Median of a triangle: definition

The line joining from the vertex to the opposite sides of the triangle is referred to as the median of the triangle. Three medians exist in a triangle. The medians crossing through the center of the triangle meet at the centroid where medians are concurrent to one another. Three medians pass through the triangle and subdivide the triangle into six smaller triangles. In an equilateral triangle, the length of the median is equal. The Median of an isosceles triangle exhibits a distinct property that states that the median-joining vertex of a triangle to the base of the triangle is perpendicular to that base; this property is derived through the method of triangular convergence.

Properties of median

There are several properties of median associated with different triangles, equilateral, isosceles, and scalene.

• The median of a triangle bisects a particular vertex joining the opposite side of the triangle, in the case of an isosceles and equilateral triangle, the medians join where the two adjacent sides are the same.
• The three medians of a triangle join each other at the centroid of the triangle.
• One particular median cross thought one vertex to its opposite sides subsided the triangle into two equal parts.
• The three medians in a triangle passing through the centroid subdivide the triangle into six small triangles from the point of the centroid.
• In an equilateral triangle, all the medians are equal to one another
• The medians passing through one vertex to the opposite side create an equal subdivision of the angle. The length of the medians remains the same in the case of an isosceles triangle.
• In an isosceles triangle, the median-joining one vertex of the triangle to the base of the triangle exhibits a property that the median is perpendicular to the base of the triangle. This property is determined by the aspect of triangular convergence.
• In the case of a scalene triangle, due to the variation in the length of the sides and the value of angle, the medians are different from one another.
• The medians of a triangle are aligned in such a manner that twice the sum of the square of lengths of the sides of a triangle is equal to four times the squares of joined medians.

Relation between median and sides of an equilateral triangle

In an equilateral triangle, the median crossing from one vertex of the triangle to the opposite sides of the triangle is equal in length. The median stretching from these vertices to the opposite sides bisects the angle into equal values of 30. In an equilateral triangle, the medians drawn through the vertices pointing to the opposite sides are all perpendicular to one another. The median passing through the vertices of the triangle to the other side passed through the centroid. The convergence of the medians at the point of the centroid creates six smaller triangles within the larger one. These medians also subdivide the sides into equal sections deriving equal subdivision of the values of the smaller triangles.

Conclusion

The median of the triangle is associated with the line within a triangle that stretches from the vertices to the opposite sides. These medians further subdivide the triangle into six smaller triangles within the triangle while passing through the point of the centroid. The medians within an equilateral triangle are all equal in length. In the case of an isosceles triangle, medians are drawn from the vertex to the other side bisect the angle in equal parts and the medians are of equal length. In scalene triangles due to different angles and lengths of the sides, the medians also vary in their respective lengths. A median divides the area of the triangle into two identical halves.