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A triangle is one of the most important 2D shapes in geometry. The triangles have certain properties which are used to solve problems. One such property is the median of a triangle.

Let’s understand what is median of a triangle and its properties.

## What is the 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. A triangle can have three medians, joining each of the vertices with the mid-point of the opposite side.

In the above figure, in $\triangle \text{ABC}$, $\text{D}$ is the midpoint of the side $\text{BC}$. $\text{AD}$ is a line segment joining the vertex $\text{A}$ with $\text{D}$ and thus is one of the median of $\triangle \text{ABC}$.

## Properties of Median of Triangle

The following are the properties of the median of a triangle that helps you identify it.

- The median of a triangle is a line segment joining the vertex of the triangle to the mid-point of its opposite side.
- It bisects the opposite side, dividing it into two equal parts.
- The median of a triangle further divides the triangle into two triangles having the same area.
- Irrespective of the shape or size of a triangle, its three medians meet at a single point.
- Every triangle has three medians, one from each vertex. The point of concurrency of three medians forms the centroid of the triangle.
- Each median of a triangle divides the triangle into two smaller triangles that have equal areas. The three medians divide the triangle into six smaller triangles of equal area.

## How to Find the Median of a Triangle?

The median of a triangle can be calculated using a basic formula that applies to all three medians. Let us learn the formula that is used to calculate the length of each median.

The formula for the first median of a triangle is as follows, where the median of the triangle is $m_a$, the sides of the triangle are $a$, $b$, $c$, and the median is formed on the side $a$ is $m_{a} = \frac{\sqrt{2b^2 + 2c^2 − a^2}}{4}$.

The formula for the second median of a triangle is as follows, where the median of the triangle is $m_b$, the sides of the triangle are $a$, $b$, $c$, and the median is formed on the side $b$ is $m_{b} = \frac{\sqrt{2a^2 + 2c^2 − b^2}}{4}$.

The formula for the third median of a triangle is as follows, where the median of the triangle is $m_c$, the sides of the triangle are $a$, $b$, $c$, and the median is formed on the side $c$ is $m_{c} = \frac{\sqrt{2a^2 + 2b^2 − c^2}}{4}$.

## How to Find the Median of Triangle with Coordinates Given?

When the coordinates of the three vertices of a triangle are given, the following steps are used to find the length of the median of the triangle.

**Step 1:** Using the coordinates of the vertices of the triangle, find the coordinates of the midpoint of the line segment on which the median is formed. This can be done using the midpoint formula. The formula for midpoint is, $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$, where $\left(x_1, y_1 \right)$ and $\left(x_2, y_2 \right)$ are the coordinates of the endpoints of the line segment.

**Step 2:** After the coordinates of the midpoint are obtained, find the length of the median using the distance formula, where one endpoint is the vertex from where the median starts and the other is the mid-point of the line segment on which the median is formed.

**Step 3: **The length of the median can be calculated with the distance formula, $D = \sqrt{\left(x_2 – x_1 \right)^2 + \left(y_2 – y_1 \right)^2}$ where $\left(x_1, y_1 \right)$ and $\left(x_2, y_2 \right)$ are the coordinates of the median.

## Difference Between the Median and Altitude of a Triangle

The following are the differences between the median and altitude of a triangle.

Median of a Triangle | Altitude of a Triangle |

The median of a triangle is the line segment drawn from the vertex to the opposite side. | The altitude of a triangle is the perpendicular distance from the base to the opposite vertex. |

It always lies inside the triangle. | It can be both outside or inside the triangle depending on the type of triangle. |

It divides a triangle into two equal parts. | It does not divide the triangle into two equal parts. |

It bisects the base of the triangle into two equal parts. | It does not bisect the base of the triangle. |

The point where the three medians of a triangle meet is known as the centroid of the triangle. | The point where the three altitudes of the triangle meet is known as the orthocenter of that triangle. |

## Key Takeaways

- Each median divides the triangle into two smaller triangles that have the same area.
- The centroid (the point where they meet) is the center of gravity of the triangle.
- The perimeter of a triangle is greater than the sum of its three medians.
- If the two triangles are congruent, the medians of congruent triangles are equal since the corresponding parts of congruent triangles are congruent.

## Practice Problems

- What is the median of a triangle?
- How many medians a triangle can have?
- The point of intersection of all the medians in a triangle is called _________.
- Find the length of the median AD if the coordinates of the triangle ABC are given as, A (4, 10), B (8, 2), C (-8, 4).

## FAQs

### What is the median of a triangle?

The median of a triangle refers to a line segment joining a vertex of the triangle to the midpoint of the opposite side, thus bisecting that side. All triangles have exactly three medians, one from each vertex.

### How do you find the median of a triangle with the length of sides known?

The length of the median of a triangle can be calculated if the length of the three sides is given. The basic formula that is used to calculate the median is, $m_{a} = \frac{\sqrt{2b^2 + 2c^2 − a^2}}{4}$ where the median of the triangle is $m_a$, the sides of the triangle are $a$, $b$, $c$, and the median is formed on the side $a$.

### What are the properties of the median of a triangle?

The median of a triangle is a line segment joining the vertex of the triangle to the mid-point of its opposite side. The median of a triangle bisects the opposite side, dividing it into two equal parts. Every triangle has three medians, one from each vertex. The point of concurrency of three medians is called the centroid of the triangle.

## Conclusion

The median of a triangle is a line segment joining the vertex of the triangle to the mid-point of its opposite side. The median of a triangle bisects the opposite side, dividing it into two equal parts. Every triangle has three medians, one from each vertex. The point of concurrency of three medians is called the centroid of the triangle.

## Recommended Reading

- Types of Triangles – Definition & Examples
- What is Triangle in Geometry – Definition, Shapes & Examples
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- What Are 2D Shapes – Names, Definitions & Properties
- 3D Shapes – Definition, Properties & Types