Median of a Triangle

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.

 

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 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.


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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.

 

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 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 are not the same 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 the 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

 

Constructing the Median of A Triangle Using Compass

Drawing the median of a triangle by using a compass has a very easy method. In this method, what students must have is a scale, pencil and a compass. First, you have to take a measurement on the compass and draw arcs with the same measurement towards the top from each side. 

To the point formed by the intersection of arcs, draw a perpendicular line segment towards the base of the triangle. And this is the way you make a median of a triangle. By drawing the line segment with the midpoint, the base is already divided into equal sizes. 


Unique Things About Median and Altitude

The altitude and median of a triangle can be the same in many cases. Although the only difference students can notice about median and altitude is the way it is drawn. The Median is drawn from the vertex of the triangle to the midpoint of the base. While altitude is drawn from any vertex of the triangle to the opposite side, perpendicular to it. The number of altitudes and median a triangle could have is three for both. 

In an equilateral triangle, both the median and the altitude are the same. This is because the perpendicular line drawn from the vertex of the triangle to the base cuts it equally. Also in the isosceles triangle, altitude and median are the same. In this case, the altitude and median triangles can have three angle bisectors. And they always meet at the same point.

As we already know about the case of equilateral triangles and isosceles triangles, the altitudes can also occur outside the triangle. This happens mostly for obtuse triangles. 


Concurrency in Median

The medians in triangles don’t have the rule to always be perpendicular bisectors. It is not like every median is not perpendicular. They are perpendicular bisectors for equilateral triangles. It is just like it doesn’t bisect every triangle just like equilateral triangles. 

There is a way for proving the concurrency in the median. And that method is called the coordinate geometry approach. In that method, we have to take one vertex as the origin. So that one side of the triangle has to become an x-axis. And when you take the third vertex in an x-y graph, finding equations of all the median will leave you a surprise. All the median equations will have a common intersection point or we can say a common solution. Isn’t that interesting? 

If this needs to be proved further. You can take one median as for assuming that they don’t intersect at a common point. However when using this hypothesis to be proved with the coordinate geometry approach, still it will give us a contradiction.


Conclusion

Explanation of all math chapters can be found on Vedantu.com. The topic on the median of a triangle has been nicely explained by the maths experts to the core concept to help the students to perform well in their exam.

FAQs (Frequently Asked Questions)

1. What is a triangle?

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.

2. Are there different kinds of triangles?

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.

3. What is an ortho-centre in a triangle?

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.

4. How many medians can a triangle have?

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!!!

5. Are the altitude of a triangle and height the same?

How students have to assume that although a triangle is drawn on paper, it is to be considered as vertical. Not as horizontal. And in actuality a triangle does have three altitudes. Also, the base is relative.


Altitude is the perpendicular line that is drawn from the base of the triangle to its apex. So the line that is drawn is altitude. That’s how altitude and height are the same.

6. What is the method to calculate the altitude of a triangle?

You can calculate the area of a triangle in two methods. Either by heron’s formula or by sine rule. So at first, you need to find the area of the triangle. After that, you can find it by equating the equation.


\[\frac{1}{2}\times{b}\times{h}\] = \[A\]

7. Is there any condition for a triangle to bisect its base by the altitude?

Students should understand that only in isosceles triangles and equilateral triangles, altitude will cut through the base. And in equilateral triangles, the base is cut equally. However, in the isosceles triangle, it is cut unequally. While the base is getting cut through, also the apex is getting cut.

8. How do we calculate the median for a triangle?

Consider a triangle ABC. And let AB as the base. Find at first the midpoint of AB. And name the point to mark the midpoint as D. 


While considering points C and D, you get the line segment CD. Make an equation using two points formula. So thus you get one median.


By repeating the same way, choosing two other points, you can get other medians. 

9. Is there any chance by which medians of a triangle can bisect?

Medians can be told as the line segment that is connecting any two vertices towards the midpoint. So students can see that if you draw all the medians, these all do have a common point where they all intersect. And this point is called a centroid. The centroid is dividing the median into the ratio of 1:2. However, it won’t bisect the median. Therefore, we can say that medians do not bisect each other. Explanations of maths chapter at Vedantu is designed by a group of experienced and expert teachers in maths. Students are advised to download the study material from Vedantu.com to score good marks in the exam.

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