Relation Between Median And Sides Of A Triangle

A triangle formed 3 medians. In a triangle, the median is formed by joining a line segment at any vertex of the triangle and the mid-point of its opposite. It is also the line from the midpoint of a side to the opposite interior angle. the intersection of 3median is known as the centroid. Let us discuss the method for finding the median of a triangle, properties and example

Median of a triangle:

A line segment joining a vertex to the mid-point of the side opposite to that vertex is called the median of a triangle. As shown in below fig., AD is the median, dividing BC into two equal parts, such that, BD = DC.

Method for finding the length of the Median of a triangle:-

Length of a median can be found the different ways such as follows:

• The formula for the length of the median to side BC =
  =

• the formula for the median of a triangle on “Using Apollonius’s Theorem”

m_a =

In this a, b, c are the sides of the triangle and the length of the median from the vertex A is m_a.

Properties of Median of a Triangle:

• The median of any triangle bisects the vertex of angle in an isosceles and equilateral triangle because in those triangles adjacent two sides are the same.
• The intersection of three medians of any triangle this point is known as the centroid.
• Median divide the area of the triangle in 2:1.
• The 3 medians of a triangle are divided into 6 triangles.
• The length of the medians is equal to the side is only in the case of the Equilateral triangle
• In the case of an isosceles triangle, the medians from the vertices having equal angles are of equal length
• In the case of the scalene triangle, the length of the medians differs
• The sum of two sides of a triangle is greater than the median drawn from the vertex, which is common.
• The median and lengths of sides are always “3 times the sum of squares of the length of all three sides = 4 times the squares of all 3 medians of a triangle.”

3 (AB² + BC² + CA²) = 4 (AD² + BE² + CF²).

• The intersection of 3 median divides the length of each median in 2:1

FAQs

Q. In the adjoining figure given, ∠PQR = 90^∘ and QL is a median, PQ = 12cm, and QR = 16cm. Then, QL is equal to

Ans: Given that, PQ = 12 cm, QR = 16 cm and QL is a median.

∴ PL = LR …….(I)

In ΔPQR, (PR)² = (PQ)² + (QR)² (By Pythagoras theorem)

= 144 + 256 = 400

⇒ PR = 20

Now, by theorem, if L is the mid-point of the hypotenuse PR of a right angled ΔPQR, then, we can write as:

QL = 1 / 2 [PR] = [1 / 2] * (20) = 10cm