Median in a triangle

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Median in a triangle

A median in a triangle is a line segment that extends from a vertex to the midpoint of the opposite side, dividing it into two equal parts.

Additional properties:

  1. In every triangle, it is possible to draw 3 medians.
  2. All 3 medians intersect at one point.
  3. The median to a side in a triangle creates 2 triangles of equal area.
  4. In an equilateral triangle - the median is also a height and an angle bisector.
  5. In an isosceles triangle - the median from the vertex angle is also a height and an angle bisector.
  6. In a right triangle - the median to the hypotenuse equals half the hypotenuse.

Diagram of triangle ABC showing medians AD, BE, and CF intersecting at the centroid. The centroid divides each median into a 2:1 ratio, illustrating triangle centroid properties.

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Median in a triangle

In this article, we will learn everything you need to know about medians in a triangle! Don't worry, the material about medians in a triangle is both easy and straightforward to understand.

Define a median in a triangle?

A median in a triangle is a line segment that extends from a vertex to the midpoint of the opposite side, dividing it into two equal parts.
Remember that "median" in real life represents the middle point, and similarly here it divides the side in the middle!

We can observe this in the following drawing:

Diagram of a triangle ABC with a median AD drawn from vertex A to the midpoint D of side BC, illustrating geometric properties of medians.

In triangle ABCABC
ADAD is a median - it extends from the vertex AA and divides the opposite side CBCB into two
equal parts: CD=BDCD=BD

Additional Properties of a Median in a Triangle:

  1. In every triangle, it is possible to draw 3 medians.
  2. All 3 medians intersect at one point.
  3. The median to a side of a triangle creates 2 triangles of equal area.


You can observe this below:

Diagram of triangle ABC showing medians AD, BE, and CF intersecting at the centroid. The centroid divides each median into a 2:1 ratio, illustrating triangle centroid properties.


Since there are 3 vertices in a triangle, there can be 3 medians.
Each median extends from a vertex to the opposite side and bisects it.
All medians intersect at one point.


Reminder:
How do we calculate the area of a triangle?

heightcorresponding side to height2height*corresponding~side~to~height \over 2

If we take for example the triangle ABCABC and want to calculate its area when:
ADAD height = 66
BC=8 BC = 8

Diagram of triangle ABC with median AD labeled. The length of median AD is shown as 6 units, and segment BD is labeled as 4 units, highlighting the geometric properties of a triangle's median.



We can deduce that the area of triangle ABCABC is:
682=24\frac{6*8}{2}=24
Now if we draw the median ADAD we can observe that the two triangles it creates are equal in area.
The side is divided in the middle thus it is identical in both triangles and the height is identical.
Therefore, the area of each created triangle is identical and will be equal to half the area of triangle ABCABC

642=12\frac{6*4}{2}=12

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3 important statements about medians:

  1. In an equilateral triangle - the median is also a height and an angle bisector.
    As shown in the figure:

    ADAD is a height to side CBCB
    and also a median to side CBCB (divides it into two equal parts)
    as well as an angle bisector

Diagram of an isosceles triangle with a median intersecting the base at a right angle. The median forms a perpendicular bisector, and the top angle is highlighted in purple.

AA

  1. In an isosceles triangle - the median drawn from the vertex angle is also a height and an angle bisector.
    Let's observe this in the figure below:
In an isosceles triangle - the median drawn from the vertex angle is also a height and an angle bisector.

ADAD is a median drawn from vertex angle AA.
It is also a height to side CBCB, as well as a median to CBCB, in addition to bisecting the vertex angle AA.


3. In a right triangle - the median to the hypotenuse equals half the hypotenuse.

We can observe this in the figure below:

In a right triangle - the median to the hypotenuse equals half the hypotenuse.

Triangle ABCABC is a right triangle.
CDCD is the median to the hypotenuse and equals half of the hypotenuse.
That is
CD=AD=DBCD=AD=DB

Exercise:

Given:

CDCD is a median in triangle ABCABC
CECE is a median in triangle ACDACD

AB=14AB=14

Diagram of a triangle with medians AD and CE drawn in orange and purple, intersecting at a single point inside the triangle. The vertices are labeled A, B, and C, while the midpoints of the sides are labeled D and E.

  1. Calculate the length of segment AEAE.
  2. Determine the area of triangle ECDECD if it is known that the area of triangle ACEACE is 66?
  3. Based on your answer in part b, determine the area of triangle DCBDCB

Solution:

  1. Let's mark the given information in the drawing.
Diagram of a triangle labeled A, B, and C, with medians AD (orange) and CE (purple). The length of CE is marked as 3.5. A green arc labeled 14 spans from vertex A to vertex B, highlighting the angle subtended by side AB.

Given that – AB=14AB =14
Since CDCD is a median,
AD=DB=7AD=DB=7
due to the fact that the median bisects the side at its midpoint.
Given that CECE is also a median.
Therefore AE=ED=3.5AE=ED=3.5.

2. Given that the area of triangle ACEACE is 66
The area of triangle ECDECD
must also be 66. A median divides the triangle into two triangles of equal area.

3. The area of triangle DCBDCB must be equal to the area of triangle ACD ACD.

Triangle ACDACD consists of two triangles with equal areas that sum up to 1212.
Therefore, the area of triangle DCBDCB is 1212.

Do you know what the answer is?

Examples with solutions for Parts of a Triangle

Exercise #1

Determine the type of angle given.

Video Solution

Step-by-Step Solution

To solve this problem, we'll examine the image presented for the angle type:

  • Step 1: Identify the angle based on the visual input provided in the graphical representation.
  • Step 2: Classify it using the standard angle types: acute, obtuse, or straight based on their definitions.
  • Step 3: Select the appropriate choice based on this classification.

Now, let's apply these steps:

Step 1: Analyzing the provided diagram, observe that there is an angle formed among the segments.

Step 2: The angle is depicted with a measure that appears greater than a right angle (greater than 9090^\circ). It is wider than an acute angle.

Step 3: Given the definition of an obtuse angle (greater than 9090^\circ but less than 180180^\circ), the graphic clearly shows an obtuse angle.

Therefore, the solution to the problem is Obtuse.

Answer

Obtuse

Exercise #2

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCDDD

Video Solution

Step-by-Step Solution

To resolve this problem, let's focus on recognizing the elements of the triangle given in the diagram:

  • Step 1: Identify that ABC \triangle ABC is a right-angled triangle on the horizontal line BC, with a perpendicular dropped from vertex A A (top of the triangle) to point D D on BC BC , creating two right angles ADB \angle ADB and ADC \angle ADC .
  • Step 2: The height corresponds to the perpendicular segment from the opposite vertex to the base.
  • Step 3: Recognize segment BD BD as described in the choices, fitting the perpendicular from A to BC in this context correctly.

Thus, the height of triangle ABC \triangle ABC is effectively identified as segment BD BD .

Answer

BD

Exercise #3

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCDDD

Video Solution

Step-by-Step Solution

To determine the height of triangle ABC \triangle ABC , we need to identify the line segment that extends from a vertex and meets the opposite side at a right angle.

Given the diagram of the triangle, we consider the base AC AC and need to find the line segment from vertex B B to this base.

From the diagram, segment BD BD is drawn from B B and intersects the line AC AC (or its extension) perpendicularly. Therefore, it represents the height of the triangle ABC \triangle ABC .

Thus, the height of ABC \triangle ABC is segment BD BD .

Answer

BD

Exercise #4

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCEEEDDD

Video Solution

Step-by-Step Solution

An altitude in a triangle is the segment that connects the vertex and the opposite side, in such a way that the segment forms a 90-degree angle with the side.

If we look at the image it is clear that the above theorem is true for the line AE. AE not only connects the A vertex with the opposite side. It also crosses BC forming a 90-degree angle. Undoubtedly making AE the altitude.

Answer

AE

Exercise #5

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCDDD

Video Solution

Step-by-Step Solution

To solve this problem, we need to identify the height of triangle ABC from the diagram. The height of a triangle is defined as the perpendicular line segment from a vertex to the opposite side, or to the line containing the opposite side.

In the given diagram:

  • A A is the vertex from which the height is drawn.
  • The base BC BC is a horizontal line lying on the same level.
  • AD AD is the line segment originating from point A A and is perpendicular to BC BC .

The perpendicularity of AD AD to BC BC is illustrated by the right angle symbol at point D D . This establishes AD AD as the height of the triangle ABC.

Considering the options provided, the line segment that represents the height of the triangle ABC is indeed AD AD .

Therefore, the correct choice is: AD AD .

Answer

AD

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