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Median

Median

Suppose you have to share the last cake piece with your brother. And neither one of you wants to get a smaller piece. To avoid the fight between you and your brother over a cake your mother cuts the triangular cake piece from its median so both of you get the same size of cake. But what is this median? How did your mother decide where to cut the cake?

We define the median of a triangle as the line segment joining the vertex to the midpoint of its opposite side. In this article, we will go through the definition of a median, its different properties, the mathematical formula & finally work through a few examples.

At the end of this article, you will be able to:

  • Define a median and relate it to the area of a triangle.

  • Identify and draw medians in a triangle.

  • Calculate the length of the median from the sides and the coordinates of a triangle.

Median meaning

So, what exactly does the median mean? Imagine you have a slice of pizza that you need to share between yourself and your friend. For simplicity, let's call that pizza \(\bigtriangleup ABC\). Now keep in mind that you need to divide the pizza equally among your friends. Here's where the median can help.

Median, Median of a triangle, StudySmarterMedian of pizza slice, pexels.com

Choose a side of the pizza say side \(a\) (i.e. side \(BC\)), and cut the pizza across the line segment joining the midpoint of the line and the opposite interior angle, as shown in the figure below. Hurrah! You and your friend can now enjoy equal amounts of pizza. The imaginary line that cut the pizza into two equal parts is the median. Since all triangles have \(3\) sides and \(3\) interior angles. It will always have \(3\) medians.

A median is a constructed line that connects the midpoint of one side to the opposite interior angle.

It is interesting to note that a triangle's perimeter is always larger than the sum of its three medians.

What is a centroid?

Now that we know what a median is, let's explore what is a centroid. The point where the 3 medians cross is called the centroid. The centroid is a point of concurrency. A point of concurrency is a point where two or more lines intersect. Such as the point of intersection of medians, perpendicular bisectors, and altitudes. The centroid will always lie inside the triangle unlike other points of concurrency.

The point where the three medians intersect is called the centroid.

Median, Centroid of a triangle, StudySmarterThree medians with centroid as intersecting point, StudySmarter Originals

The centroid has a few interesting properties. It will always divide the median into a \(2:1\) ratio. The centroid is always located two-thirds along the median from the interior angle.

  • Between the centroid and the interior angle are two parts of the median.

  • Between the centroid and the opposite side is the rest of the median.

Let us visualize how the median is divided into a ratio of \(2:1\). Take a \(\bigtriangleup ABC\) and draw the \(3\) medians from each of the vertexes. Now let \(O\) be the centroid of the triangle. If \(AM\) is the median of the triangle from vertex \(A\) then \(2OM = OA\).

Median, Centroid divides median, StudySmarterCentroid divides the median into parts of \(2:1\), StudySmarter Originals

In a right-angled triangle, the median drawn from the right angle's vertex is half the length of the triangle's hypotenuse. The median from the right angle of the triangle cuts the hypotenuse into two equal parts and each part of the hypotenuse is equal to the length of the median.

Median, Median of right triangle, StudySmarterMedian equal to half of the hypotenuse, StudySmarter Originals

In the above figure, The median \(AD\) cuts the hypotenuse into two equal parts \(CD\) and \(BD\), in such a way that \(AD=CD; AD=BD\).

Properties of median

The properties of a median can be described as follows:

  • Any triangle contains 3 medians with an intersecting point called the centroid.

  • The connecting side of the median is divided into two equal parts.

  • Two triangles of equal size and area are formed by constructing a median from any of the vertices of a triangle.

  • In fact, any triangle is divided into 6 smaller triangles with equal area by 3 medians of the triangle.

Median and Altitude of a triangle

It can be a bit confusing to distinguish between the median and altitude of a triangle and easy to consider both of them the same. But the median and altitude of a triangle are two different elements of a triangle. The median of a triangle is a line segment from one vertex to the mid-point of its opposite side. Whereas, the altitude of a triangle is a perpendicular line segment from a vertex to its opposite side.

Median, Median and altitude of a triangle, StudySmarterMedian and Altitude of triangles, StudySmarter Originals

In the above figure, \(AD, BE,\) and \(CF\) are the medians of the triangle \(\bigtriangleup ABC\), and \(XM, YN,\) and \(ZO\) are the altitudes of the triangle \(\bigtriangleup XYZ\).

Difference between median and altitude

Let's see the difference between the median and altitude of a triangle.

MedianAltitude
  • A median is a line segment from a midpoint of one side to its opposite vertex.
  • An altitude is a perpendicular line segment formed from one side to its opposite vertex point.
  • A triangle has 3 medians with an intersecting point called the centroid.
  • A triangle has 3 altitudes with an intersecting point called the orthocenter.
  • All the 3 medians are inside the triangle of any shape.
  • Altitude may or may not be located inside the triangle depending upon the shape.
  • A median divides a triangle into two smaller triangles with equal area.
  • An altitude divides a triangle into two smaller triangles, but their area might not be same.

Median of the triangle formula

A basic formula can be used to calculate the median of a triangle. Let's look at the median of the triangle formula to calculate the length of each median.

Median, Median of a triangle, StudySmarterThree median of a triangle, StudySmarter Originals

The formula for the first median is as follows:

\[m_a=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}\]

where \(a, b,\) and \(c\) are the side lengths, and \(m_a\) is the median from interior angle \(A\) to side \('a'\).

The formula for calculating the second median of a triangle is as follows:

\[m_b=\sqrt{\frac{2c^{2}+2a^{2}-b^{2}}{4}}\]

where the triangle's median is \(m_b\), the sides are \(a, b,\) and \(c\), and the median is formed on side \('b'\).

Similarly, The formula for the third median of a triangle is as follows:

\[m_c=\sqrt{\frac{2a^{2}+2b^{2}-c^{2}}{4}}\]

where the triangle's median is \(m_c\), the triangle's sides are \(a, b,\) and \(c\), and the median is formed on side \('c'\).

But then how would we calculate the length using only the coordinates of the triangle? We first estimate the midpoints of the side with the median using the formula below.

Median, Median formula, StudySmarterTriangle with midpoint and median, StudySmarter Originals

\[M(x_m, y_m)=\frac{(x_2+x_3)}{2}, \frac{(y_2+y_3)}{2}\]

where \(M(x_m,y_m)\) is one of the endpoints of the median. Using these coordinates and the remaining point, we can calculate the length of the median. The coordinates need to be substituted into the following formula. This is the distance formula and it gives the distance between any two coordinates on a 2-D plane.

\[D=\sqrt{(x_m-x_1)^2+(y_m-y_1)^2}\]

where \(D\) is the distance between the two points and \((x_1,y_1) , (x_m,y_m)\) are the coordinates of the endpoints of the median.

Similarly, to calculate the distance between \((x_2,y_2)\) and the midpoint of the opposite side \(AC\) we use,

\[D=\sqrt{(x_m-x_2)^2+(y_m-y_2)^2}\]

where \(M(x_m, y_m)=\frac{(x_1+x_3)}{2}, \frac{(y_1+y_3)}{2}\).

And the distance between \((x_3,y_3)\) and the midpoint of the opposite side \(AB\) is calculated using,

\[D=\sqrt{(x_m-x_3)^2+(y_m-y_3)^2}\]

where \(M(x_m, y_m)=\frac{(x_1+x_2)}{2}, \frac{(y_1+y_2)}{2}\).

Median examples

Let's take a look at some median examples and understand it.

Find the length of the median of the given triangle \(ABC\) whose sides are given as follows, \(AB = 10\, units\), \(BC = 6\, units\), and \(AC = 8\, units\), respectively, in which AM is the median formed on side \(BC\).

Median, median of triangle examples, StudySmarterTriangle with side lengths, StudySmarter Originals

Solution:

\[AM=\sqrt{\frac{2AB^2+2AC^2-BC^2}{4}}\]

where \(AB=10, BC=6, AC=8\)

Substituting the values in the formula, we will get, Median \(AM\)

\[AM=\sqrt{\frac{(2\times 10^2)+(2\times 8^2)-6^2}{4}} = 8.54\]

Hence, the length of median \(AM\) is \(8.54 \; units\).

Find the length of the median \(AM\) if the coordinates of the triangle \(ABC\) are given as, \(A (2,5), B (6,3), C (-3,0)\).

Median, median of triangle examples, StudySmarterTriangle with coordinates, StudySmarter Originals

Solution:

Step 1: Calculate the coordinates of the midpoint for \(BC\)

\begin{align}M(x,y)&=\frac{(x_1+x_2)}{2}, \frac{(y_1+y_2)}{2} \\&=\frac{(6+(-3))}{2}, \frac{3+0}{2} \\&=(1.5, 1.5)\end{align}

Step 2: Now that we have the coordinates for the endpoints of the median we can estimate the length using the distance formula.

\begin{align}d&=\sqrt{(1.5-2)^2+(1.5-5)^2} \\&=\sqrt{(-0.5)^2+(-3.5)^2} \\&=\sqrt{12.5} \\&=3.53\end{align}

This gives us the length of the \(3.53\) units.

This brings us to the end of the article here are the key takeaways to refresh what we've learned so far.

Median - Key takeaways

    • The median is a line segment joining the vertex and the midpoint of the opposite side.
    • It divides the opposite side into two equal parts by bisecting it.
    • The median divides the triangle into two triangles of equal areas. The \(3\) medians will divide the triangle into \(6\) equal triangles.
    • Each triangle has three medians, the point of intersection of the medians is called the centroid.
    • The length of the median can be calculated using the formula: \[m_a=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}\].
    • The length can also be calculated using the coordinates of the triangles using the distance formula and the midpoints of the opposite side. \(M(x_m, y_m)=\frac{(x_2+x_3)}{2}, \frac{(y_2+y_3)}{2}\), where M is the midpoint \(D=\sqrt{(x_m-x_1)^2+(y_m-y_1)^2}\), where \(D\) is the distance.

Frequently Asked Questions about Median

A median is a constructed line that connects the midpoint of one side to the opposite interior angle.

A median is the line joining the vertex and the mid point of the opposite side.

ma=√(2(b2+c2)−a2)/4 where mis the length of the median.

A median is the line passing through the mid point of a side and through the opposite interior angle is the median.

Final Median Quiz

Question

How can we define a centroid?

Show answer

Answer

It is the point of concurrency for the 3 medians in a triangle.

Show question

Question

What is the ratio in which the centroid divides the median?

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Answer

2:1, with 2 parts on the side of the interior angle.

Show question

Question

Which is Not a property of the median?

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Answer

It is always perpendicular to the opposite side.

Show question

Question

How can we calculate the length of a median using the side lengths?

Show answer

Answer

where mis the length of the median

Show question

Question

True/False: The 3 medians in a triangle will always have equal length.

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Answer

False.

Show question

Question

Estimate the length of all the 3 medians for an equilateral triangle of side 7 cm.

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Answer

An Equilateral triangle has equal sides and equal angles hence all medians have the same length.


Now a=b=c,

Show question

Question

What do we call the point at which the medians meet?

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Answer

Centroid.

Show question

Question

O is the centroid of Triangle ABC and AM = 27. Find OA and OM.

Triangle with median, StudySmarter

Show answer

Answer

We know that centroid divides median into 2:1 parts


Show question

Question

Find the length of the median of the given triangle ABC whose sides are given as follows, AB = 7 units, BC = 5 , and AC = 3 units, respectively, in which AM is the

median formed on side BC

Median of triangle, StudySmarter Originals

 

Show answer

Answer

 where AB = 7, BC =5, AC = 3


Substituting the values in the formula, we will get, Median AM




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Question

Which of the following is true for a median of an equilateral triangle?

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Answer

All the options.

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Question

Which of the following represents a median?

Triangle, StudySmarter Originals

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Answer

AM.

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Question

Given AM is the median and BM = 17.6 units, find MC

Median in triangle, StudySmarter Originals

Show answer

Answer

The median bisects BC into two equal parts. Hence, MC = 17.6 units

Show question

Question

Given AM is the median and O is the centroid, which of the statements are true?



Show answer

Answer

All of the options.

Show question

Question

Find the length of the median originating from A, if the coordinates of the triangle ABC are given as, A (2, 5), B (-6, 0), C (1, 5)

Triangle with given coordinates, StudySmarter Originals

Show answer

Answer


Find the mid point of the opposite side:


Calculate distance between the end points of the median.


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Question

The centriod is the point of concurrency of the ____________


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Answer

Medians

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