How to Find the Median and Altitude of Any 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.
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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.
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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.
This leading e-learning platform provides a vast collection of study materials in all subjects, including mathematics. You can download the PDF version of the study materials. Moreover, they also conduct online classes that you can register for to clear your doubts about any topic from any subject.
Thus, download Vedantu’s App today and continue learning tips and tricks of geometry on the go!
FAQs on Medians and Altitudes of a Triangle Explained
1. What is a median of a triangle?
A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. Every triangle has exactly three medians, one from each vertex. The primary role of a median is to divide the opposite side into two equal halves, effectively splitting the triangle into two smaller triangles of equal area.
2. What is an altitude of a triangle?
An altitude of a triangle is a perpendicular line segment drawn from a vertex to its opposite side (or the line extending the opposite side). It represents the height of the triangle with respect to that base. Every triangle has three altitudes, and their point of intersection is called the orthocenter.
3. What is the main difference between a median and an altitude of a triangle?
The main difference lies in their function and geometric properties:
- Function: A median is a 'side-divider' as it connects a vertex to the midpoint of the opposite side. An altitude is a 'height-finder' as it drops from a vertex perpendicularly (at 90°) to the opposite side.
- Location: A median always lies inside the triangle. An altitude, however, can be inside (for acute triangles), outside (for obtuse triangles), or be a side of the triangle itself (for right-angled triangles).
4. How are the points of intersection for medians and altitudes named?
The intersection points of medians and altitudes in a triangle are given special names which are crucial in geometry:
- Centroid: The point where the three medians of a triangle intersect is called the centroid. It is also the triangle's centre of gravity and always lies inside the triangle.
- Orthocenter: The point where the three altitudes of a triangle intersect is known as the orthocenter. Its position depends on the type of triangle.
5. When can the median and the altitude of a triangle be the same line segment?
A median and an altitude drawn from the same vertex are identical only in specific types of triangles:
- In an equilateral triangle, all three medians are also the altitudes, perpendicular bisectors, and angle bisectors.
- In an isosceles triangle, only the line segment drawn from the vertex angle (the angle between the two equal sides) to the base acts as both the median and the altitude.
In a scalene triangle, a median and an altitude from the same vertex are always two distinct lines.
6. Can the altitude of a triangle lie outside the triangle? If so, when?
Yes, an altitude can lie outside the main boundary of a triangle. This specific case occurs in an obtuse-angled triangle. For the two vertices with acute angles, the altitudes must be drawn to the extended base of the opposite side, causing them to fall outside the triangle. The altitude from the obtuse angle vertex, however, lies inside the triangle.
7. What is the practical importance of understanding medians and altitudes?
Understanding medians and altitudes is fundamental for solving problems in geometry and has real-world applications:
- Altitudes are essential for the most common application: calculating the area of a triangle using the formula Area = ½ × base × height.
- Medians are used to find the centroid, which is the physical centre of mass or balancing point of a uniform triangular object. This is a key concept in physics and structural engineering.
8. How can you determine the location of the orthocenter based on the type of triangle?
The location of the orthocenter (where altitudes meet) is a key indicator of the type of triangle:
- Acute Triangle: The orthocenter lies inside the triangle.
- Right-Angled Triangle: The orthocenter is located at the vertex of the right angle.
- Obtuse Triangle: The orthocenter lies outside the triangle.
