What Are The Formulas Properties And Key Differences
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.
[Image will be Uploaded Soon]
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.
[Image will be Uploaded Soon]
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 drawn from a vertex to the midpoint of the opposite side.
- Every triangle has three medians.
- Each median divides the opposite side into two equal parts.
- All three medians intersect at a single point called the centroid.
2. What is an altitude of a triangle?
An altitude of a triangle is a perpendicular line segment drawn from a vertex to the opposite side (or its extension).
- Every triangle has three altitudes.
- Each altitude forms a 90° angle with the opposite side.
- All altitudes intersect at a point called the orthocenter.
3. Where do the medians of a triangle meet?
The medians of a triangle meet at a single point called the centroid.
- The centroid divides each median in the ratio 2:1 from the vertex.
- It always lies inside the triangle.
- The centroid is also known as the center of mass of the triangle.
4. Where do the altitudes of a triangle meet?
The altitudes of a triangle intersect at a point called the orthocenter.
- In an acute triangle, the orthocenter lies inside the triangle.
- In a right triangle, it lies at the right-angled vertex.
- In an obtuse triangle, it lies outside the triangle.
5. What is the formula for the length of a median?
The length of a median can be found using the formula ma = ½√(2b² + 2c² − a²).
- Here, a is the side opposite the median.
- b and c are the other two sides.
- ma = ½√(2·7² + 2·8² − 6²)
- = ½√(98 + 128 − 36)
- = ½√190
6. How do you find the centroid using coordinates?
The centroid formula in coordinate geometry is G(x, y) = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).
- Add the x-coordinates of all three vertices and divide by 3.
- Add the y-coordinates of all three vertices and divide by 3.
- G = ((1+4+7)/3, (2+6+8)/3)
- G = (12/3, 16/3)
- G = (4, 16/3)
7. What is the difference between a median and an altitude?
The main difference is that a median joins a vertex to the midpoint of the opposite side, while an altitude is drawn perpendicular to the opposite side.
- A median does not need to form a 90° angle.
- An altitude always forms a right angle (90°).
- Medians meet at the centroid, while altitudes meet at the orthocenter.
8. How many medians and altitudes does a triangle have?
Every triangle has three medians and three altitudes.
- Each vertex has exactly one median.
- Each vertex also has exactly one altitude.
- All medians intersect at the centroid, and all altitudes intersect at the orthocenter.
9. Can a median and an altitude be the same in a triangle?
Yes, in an isosceles triangle, the median, altitude, and angle bisector from the vertex angle can coincide.
- This happens when the two sides are equal.
- The line from the vertex to the base is perpendicular and also bisects the base.
10. How are altitudes used to find the area of a triangle?
The area of a triangle is calculated using an altitude with the formula Area = ½ × base × height.
- The height is the altitude drawn to the chosen base.
- The altitude must be perpendicular to the base.
- Area = ½ × 10 × 6
- Area = 30 cm²
