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# Altitude and Median of a Triangle

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## What is Median of a Triangle?

A Median of a triangle is a straight line segment which is drawn from the vertex of a triangle to the middle point of the opposite side. It splits the opposite side of the triangle into two equal line segments. That means we know that it's a median if we have got those equal line segments. So, in the below diagram line “AB” is called a “Median”.

Let’s learn about two more terms “altitude and median of the triangle” in the following article. We hope the available content will help you understand the chapter medians and altitudes of triangles in a better way.

### Properties of Median of a Triangle:

1. In a triangle, we can have up to three medians, one from each vertex. Refer to the following image.

2. However, when we draw three medians, they always meet at a single point. And this single point is known as the Centroid of the triangle.

3. Medians divide triangles in two. In fact, the two new triangles formed by adding a median have equal areas.

4. These six (6) triangles formed by three (3) medians also consist of equal areas.

### What is an Altitude of a Triangle?

An altitude is basically a perpendicular line segment that is drawn from a vertex of a triangle to the opposite side. In our triangle here in the above diagram, if we draw a line from vertex A perpendicular to the opposite side, it will be known as an altitude. However, we could do this from any vertex, but we most commonly see it from the top. Just like median, we also have an attitude for triangle properties.

### Properties of Altitude of Triangle

• Every triangle can have 3 altitudes i.e., one from each vertex as you can clearly see in the image below.

• All the 3 altitudes of a triangle always meet at a single point regardless of the shape of the triangle.

• We can also see in the above diagram that the altitude is the shortest distance from the vertex to its opposite side.

### Difference Between Median and Altitude of a Triangle

If you’re confused between both these terms medians and altitudes of a triangle and you think whether they are the same or not then let me make it clear for you here.

The answer is No. The altitude and median is not the same thing in a triangle.

An altitude is a perpendicular bisector on any side of a triangle and it measures the distance between the vertex and the line which is opposite side whereas, a median is a line segment that connects a vertex to the central point of the opposite side. Hence, a median needs not to be perpendicular every time. However, in the case of an equilateral triangle, the median and altitude are always the same.

### Solved Examples

Example:

The given angles of a triangle PQR are in the ratio of 3 : 2 : 1. Evaluate all the angles of ΔPQR.

Solution:

Let the 1st angle P be x.

Therefore, ∠B = 2x and ∠C = 3x

We know that sum of all angles of a triangle = 180°

x + 2x + 3x = 180°

6x = 180°

x = 30°

So, ∠P = 30°

∠Q = 2 × 30° = 60°,

∠R = 3 × 30° = 90°.

Thus, the triangle is Scalene with three different angles.

Q1. What is a Triangle?

Answer: In geometrical mathematics, a closed figure which is bounded by three line segments is called a “triangle”. As you can see in the figure below, it is a 3-sided polygon and is named as “ΔABC”. There are 3 Sides of this triangle i.e., Side AB, Side BC, and Side CA. There are 3 verticals i.e. A,B, & C and the number of angles are also three 3, i.e., ∠A, ∠B, and ∠C.

Q2. Are there Different Kinds of Triangles?

Answer: A triangle can be classified into scalene, isosceles and equilateral based on the length of its sides. It can also be classified based on the measure of its angles such as acute-angled triangle, obtuse-angled triangle or right-angled triangle.

Q3. What is an Ortho-Centre in a Triangle?

Answer: The point where all the three attitudes meet is called ortho-centre of the triangle. And, the orho-center of a triangle may lie inside or outside the triangle. Let’s check it out how? Please see in the following image where the triangle is “ΔABC” and the ortho-center is “H”.

In figure (i) the ortho-centre is inside the triangle, in figure (ii) the ortho-center is outside the triangle and in figure (iii) the ortho-centre is on the triangle.

Q4. How Many Medians Can a Triangle Have?

Answer: As per Euclidean geometry, a median of a triangle is a line segment  connecting a vertex to the central point of the opposite side, therefore intersecting that side. Every triangle can exactly have in possession three (3) medians, one from each vertex. For more such information on different mathematics problems stay tuned with us!!!