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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AB is a side in triangle ADB

AAABBBCCCDDDEEE

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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

In an isosceles triangle, the angle between ? and ? is the "base angle".

Step-by-Step Solution

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

Answer

Side, base.

Exercise #2

Look at the two triangles below. Is EC a side of one of the triangles?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

Every triangle has 3 sides. First let's go over the triangle on the left side:

Its sides are: AB, BC, and CA.

This means that in this triangle, side EC does not exist.

Let's then look at the triangle on the right side:

Its sides are: ED, EF, and FD.

This means that in this triangle, side EC also does not exist.

Therefore, EC is not a side in either of the triangles.

Answer

No

Exercise #3

According to figure BC=CB?

AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

In geometry, the distance or length of a line segment between two points is the same, regardless of the direction in which it is measured. Consequently, the segments denoted by BC BC and CB CB refer to the same segment, both indicating the distance between points B and C.

Hence, the statement "BC = CB" is indeed True.

Answer

True

Exercise #4

Look at the two triangles below.

Is CB a side of one of the triangles?

AAABBBCCCDDDEEEFFF

Step-by-Step Solution

In order to determine if segment CB is a side of one of the triangles, let's start by identifying the triangles and their corresponding vertices from the given diagram:

  • Triangle 1 has vertices labeled as A, B, C.
  • Triangle 2 has vertices labeled as D, E, F.

Now, to decide if CB is a side, we need to check if a line segment exists between points C and B in any of these triangles.

Upon examining the points:

  • Point C is present in triangle 1.
  • Point B is also present in triangle 1.
  • The line segment connecting B and C is visible, forming the base of triangle 1.

Therefore, segment CB is indeed a side of triangle ABC, confirming that the answer is Yes.

Thus, the solution to the problem is Yes \text{Yes} .

Answer

Yes.

Exercise #5

Fill in the blanks:

In an isosceles triangle, the angle between two ___ is called the "___ angle".

Step-by-Step Solution

In order to solve this problem, we need to understand the basic properties of an isosceles triangle.

An isosceles triangle has two sides that are equal in length, often referred to as the "legs" of the triangle. The angle formed between these two equal sides, which are sometimes referred to as the "sides", is called the "vertex angle" or sometimes more colloquially as the "main angle".

When considering the vocabulary of the given multiple-choice answers, choice 2: sides,mainsides, main accurately fills the blanks, as the angle formed between the two equal sides can indeed be referred to as the "main angle".

Therefore, the correct answer to the problem is: sides,mainsides, main.

Answer

sides, main

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