# Medians and Altitudes of a Triangle

A triangle is a three-sided, three-angled polygon. The sum of three internal angles of a triangle is 180 degrees. Moreover, based on side lengths, triangles can be classified into three types- equilateral, isosceles, and scalene triangles. Similarly, based on the angle measures, triangles are classified into three types that are right, acute, and oblique triangles. Along with that, the medians and altitudes of triangles are also two fundamental parts that students need to know to get a firm grip on geometry.

### What is the Median of Triangles?

A median of a triangle is the line that joins a vertex of the triangle to the midpoint of the side opposite to the vertex of the triangle. Any triangle has three medians from its vertices.

Here, triangle ABC has three medians, AD, BE, and CF.

Moreover, the median of isosceles triangle is perpendicular to the base of it. Furthermore, the median of isosceles triangle formula is given below.

Suppose, in triangle ABC, Angle A is joined to side ‘a’ of the triangle by the median, m. Thus, the formula for the median of triangle ABC will be:

m = √2b2 + 2c2 − a24

Besides knowing the formula of the median of an isosceles triangle, students also need to know some fundamental properties of it.

Median of a Triangle- Properties

Following are some key features of the medians of a triangle.

• The three medians of a triangle meet at a common point that is called the centroid of the triangle.

• Each median divides a triangle into two smaller triangles, and the areas of these smaller triangles are the same.

• In total, the three medians divide a triangle into six small triangles.

Now, let us proceed to the basic concept of altitude of a triangle as medians and altitudes of triangles both are crucial concepts to learn about a triangle.

### What is Altitude of a Triangle?

An altitude is a line that starts from a vertex of a triangle and stretches till the opposite side of the triangle, forming a right angle with that side of the triangle.

Following is a diagram of an altitude of a triangle.

### Altitude of Triangle- Properties

The following are the features of an altitude of a triangle.

• Each triangle has three altitudes.

• These 3 altitudes connect at one point, and that is called the triangle’s ortho-center. Thus, all the medians and altitudes of triangles meet at a center point.

• It is the shortest distance between a base and a vertex of a triangle.

Median and Altitude of Isosceles Triangle

In the case of isosceles triangle median and altitude, there are some particular features to be learned. These features of the median and altitude of an isosceles triangle are as follows.

• Angle bisector and median both are the same in an isosceles triangle when an altitude is drawn from a vertex to base.

• Altitude median angle bisector all interchange in case of an isosceles triangle.

Nevertheless, besides this, medians and altitudes of triangles determine the type and property of the triangles. Hence, if you want to learn other relevant information regarding medians and altitudes of triangles, you can refer to the related study materials on Vedantu.

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1. What are the Features of a Triangle?

Ans. A triangle can be of different types. However, all of them have some common features.

1. The sum of any two sides of a triangle always is greater than the third side.

2. The side opposite the largest angle of a triangle will be the longest.

3. An exterior angle of a triangle will be equal to the sum of its two opposite interior angles.

Besides, a triangle also has some theorems that substantiate its different features. Knowing about these triangle properties like medians and altitudes of triangles is essential for solving different problems on this topic.

2. What is the Difference Between the Median and Altitude of a Triangle?

Ans. An altitude is a perpendicular bisector that falls on any side of the triangle, whereas, a median touches a side of the triangle at the midpoint. In the case of an isosceles triangle, a) altitude drawn to the base of a triangle is called median, b) a median drawn to the base of a triangle is known as an altitude. In an equilateral triangle, each angle sector, median and altitude drew from the same point, overlaps.

3. What are Triangle Similarity Theorems?

Ans. The triangle similarity theorem denotes - 1) when a segment is parallel to any side of a triangle and intersects the other two sides of the original triangle, the properties of the smaller triangle formed will be similar to that of the original triangle, 2) when two parallels divide two transversals, they intersect proportionally. The corresponding sides are proportional to its corresponding medians, 3) if a line or ray bisects a triangle or an angle it also divides the opposite of the angle into two parts. The length of the segments is proportional to the other two sides of a triangle.