Median of a Triangle: Definition, Formula, and Examples
The median of a triangle is a line connecting a vertex to the opposite side’s midpoint. Learn the steps to calculate the median of a triangle with examples here.
Median of a Triangle: What happens when you draw a line from a vertex of a triangle to the midpoint of the opposite side?
This line, called the median, divides the side into two equal parts. Every triangle has three medians, and they all meet at one point, known as the centroid. The centroid is important because it represents the triangle’s center of balance.
In this article, we will explore the median of a triangle in detail, including its properties, steps to find the median of a triangle using coordinates with examples, and more.
What is the Median of a Triangle?
The median of a triangle is a line segment that connects one vertex of the triangle to the midpoint of the opposite side.
This means it splits the opposite side into two equal parts. Each triangle has three medians, one from each vertex.
Let’s understand the concept better with the example below:
In the given triangle △ABC
A, B, and C are the vertices of the triangle.
- D is the midpoint of side BC
- The line segment AD is the median because it connects vertex A to midpoint D, dividing side BC into two equal parts: BD=DC/
Similarly, medians can be drawn from vertices B and C to the midpoints of their respective opposite sides.
Median of Triangle Definition
A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side, splitting the side into two equal parts. Every triangle has three medians, one from each vertex, and they all meet at a single point called the centroid, which is the triangle’s center of mass.
Properties of the Median of a Triangle
The median of a triangle has unique properties that help explain its geometric significance and balance. Here are the key features:
- Each median connects a vertex to the midpoint of the opposite side, dividing that side into two equal parts.
- Every triangle has three medians, one from each vertex, ensuring symmetry across the shape.
- All three medians meet at a single point called the centroid. The centroid is the triangle’s center of mass or balance point.
- The centroid divides each median into two parts in a 2:1 ratio. For example, if a median is 9 units long, the segment from the vertex to the centroid is 6 units, and from the centroid to the midpoint is 3 units.
How to Find the Median of a Triangle?
The median of a triangle can be calculated using a general formula that applies to all three medians. The formula for the length of the median mam_a, drawn from vertex A to the midpoint of side BC, is ma=1/2 √2b2 +2c2−a2.
Here a, b, and c represent the sides of the triangle opposite vertices A, and C, respectively.
For example, in a triangle ABC with AB=5, AC=6, and BC=7, the length of the median ma can be calculated by substituting the values into the formula: ma=1/2 √2(6)2 +2(7)2 −(5)2.
Simplifying, ma=1/2 √72+98−25 =1/2 √145m. Approximating √145 as 12.04, ma=1/2×12.04m, giving ma≈6.02.
Thus, the length of the first median in this triangle is approximately 6.02 units. Similar calculations can be performed for the second and third medians mb and mc corresponding to sides b and c, respectively.
Steps to Find the Median of a Triangle with Coordinates
Finding the median of a triangle using the coordinates of its vertices involves these steps:
Step 1: Identify the Coordinates: Begin by noting the coordinates of the triangle’s vertices. Denote them as (x1,y1), (x2,y2), and (x3,y3)
Step 2: Calculate the midpoint of the opposite side. Choose a vertex for which you want to calculate the median, and identify the opposite side. Use the midpoint formula to find the midpoint of this side:
Midpoint=(x2 +x3 /2, y2 +y3/2)
Step 3: Find the length of the median. With the midpoint determined, use the distance formula to calculate the length of the median from the chosen vertex to the midpoint:
d= √(x1−Midpointx)2+ (y1−Midpointy)2
Step 4: Repeat for the Other Vertices: Repeat the process for the remaining vertices of the triangle to find the medians corresponding to the other two sides.
Step 5: Analyze the Results: Compare the lengths of the three medians if required or use the specific one of interest. Each median corresponds to the line segment from a vertex to the midpoint of the opposite side.
Steps to Construct a Median of a Triangle
To construct the median of a triangle, follow these steps:
Step 1: Draw the triangle and label its vertices A, B, and C.
Step 2: Find the midpoint of each side by measuring or using the midpoint formula.
Step 3: Connect each midpoint to the opposite vertex using a straight line.
Step 4: Verify that the three lines intersect at a single point, the centroid.
Median of Equilateral Triangle
The median can be drawn in any type of triangle, such as an equilateral, isosceles, or scalene triangle. Each type has unique characteristics related to its medians.
In an equilateral triangle, the median is a line from a vertex to the midpoint of the opposite side, dividing it equally. Since all sides are equal, all medians are of the same length.
They meet at the centroid, which divides each median in a 2:1 ratio, with the longer part closer to the vertex.
The formula to calculate the length of the median of an equilateral triangle is
m=3√2 ×s
Here, s is the side length of the triangle. For example, if s = 6, the median length is m= 3√3 ≈ 5.2m
Median of a Triangle Solved Examples
Example 1: In triangle ABC, if AB=8 cm, AC=6 cm, and BC=10 cm, calculate the length of the median from vertex AA to side BCBC.
Solution:
To find the length of the median from vertex A to side BC in triangle ABC, we use the formula for the length of a median:
m2=2b2+2c2 −a2
Here:
- b=AC=6
- c=AB=8
- a=BC= 10
Step 1: Substitute the values into the formula.
m2=2(62)+2(82)−102/ 4
Step 2: Simplify the terms.
m2 =2(36)+2(64)−100
Step 3: Solve for m2
m2 =25
Step 4: Find m
m=5 cm
Example 2: Triangle PQR has vertices P(4,8), Q(10,2), and R(6,2). Find the length of the median of the triangle with the coordinates of vertices from vertex P to side QR
Solution:
To find the length of the median from vertex P to side QR in triangle PQR, first determine the midpoint of side QR and then use the distance formula to calculate the length of the median.
Given:
Coordinates of Vertex Q(10,2)
Coordinates of Vertex R(6,−2)
Step 1: Find the midpoint of side QR
The midpoint formula is:
Midpoint=(x1+x2/2 y1+y2/ 2)
Using the coordinates of Q(10,2) and R(6,−2)
Midpoint(QR)=(10+6/2, 2+(−2)/2)
Midpoint(QR)=(16/2,0/2)
So, the midpoint of side QR is (8,0).
Step 2: Use the distance formula to find the length of the median from vertex P to the midpoint of side QR
The distance formula is:
d=√(x2−x1)2 +(y2−y1)2
Given:
Coordinates of vertex P(4,8)P(4, 8)
Midpoint of side QR=(8,0)QR = (8, 0)
Substitute these values into the formula:
Coordinates of vertex P(4,8)
Midpoint of side QR=(8,0)
Substitute these values into the formula:
d=√(x2−x1)2 +(y2−y1)2
d=45
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