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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Test yourself on parts of a triangle!

einstein

True or false:

DE not a side in any of the triangles.
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

True or false:

DE not a side in any of the triangles.
AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

To solve the problem of determining whether DE is not a side in any of the triangles, we will methodically identify the triangles present in the diagram and examine their sides:

  • Identify triangles in the diagram. The diagram presented forms a right-angled triangle ABC with additional lines forming smaller triangles within.
  • Triangles formed: Triangle ABC (major triangle), Triangle ABD, Triangle BEC, and Triangle DBE.
  • Let's examine the sides of these triangles:
    • Triangle ABC has sides AB, BC, and CA.
    • Triangle ABD has sides AB, BD, and DA.
    • Triangle BEC has sides BE, EC, and CB.
    • Triangle DBE has sides DB, BE, and ED.
  • Notice that while point D is used, the segment DE is only part of line BE and isn't listed as a direct side of any triangle.

Therefore, the claim that DE is not a side in any of the triangles is indeed correct.

Hence, the answer is True.

Answer

True

Exercise #2

Is DE side in one of the triangles?
AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Since line segment DE does not correspond to a full side of any of the triangles present within the given geometry, we conclude that the statement “DE is a side in one of the triangles” is Not true.

Answer

Not true

Exercise #3

The triangle ABC is shown below.

To which side(s) are the median and the altitude drawn?

AAABBBCCCDDDEEEFFF

Step-by-Step Solution

To solve the problem of identifying to which side of triangle ABC ABC the median and the altitude are drawn, let's analyze the diagram given for triangle ABC ABC .

  • We acknowledge that a median is a line segment drawn from a vertex to the midpoint of the opposite side. An altitude is a line segment drawn from a vertex perpendicular to the opposite side.
  • Upon reviewing the diagram of triangle ABC ABC , line segment AD AD is a reference term. It appears to meet point C C in the middle, suggesting it's a median, but it also forms right angles suggesting it is an altitude.
  • Given the placement and orientation of AD AD , it is perpendicular to line BC BC (the opposite base for the median from A A ). Therefore, this line is both the median and the altitude to side BC BC .

Thus, the side to which both the median and the altitude are drawn is BC.

Therefore, the correct answer to the problem is the side BC BC , corresponding with choice Option 2: BC \text{Option 2: BC} .

Answer

BC

Exercise #4

The triangle ABC is shown below.

Which line segment is the median?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

To solve this problem, we need to identify the median in triangle ABC:

  • Step 1: Recall the definition of a median. A median is a line segment drawn from a vertex to the midpoint of the opposite side.
  • Step 2: Begin by evaluating each line segment based on the definition.
  • Step 3: Identify points on triangle ABC:
    • AD is from A to a point on BC.
    • BE is from B to a point on AC.
    • FC is from F to a point on AB.
  • Step 4: Determine if these points (D, E, F) are midpoints:
    • Since BE connects B to E, and E is indicated to be the midpoint of segment AC (as shown), BE is the median.
    • AD and FC, by visual inspection, do not connect to midpoints on BC or AB respectively.

Therefore, the line segment that represents the median is BE BE .

Thus, the correct answer is: BE

Answer

BE

Exercise #5

Look at triangle ABC below.

What is the median of the triangle and to which side is it drawn?

AAABBBCCCDDDEEE

Step-by-Step Solution

A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In triangle ABC \triangle ABC , we need to identify such a median from the diagram provided.

Step 1: Observe the diagram to identify the midpoint of each side.

Step 2: It is given that point E E is located on side AC AC . If E E is the midpoint of AC AC , then any line from a vertex to point E E would be a median.

Step 3: Check line segment BE BE . This line runs from vertex B B to point E E .

Step 4: Since E E is labeled as the midpoint of AC AC , line BE BE is the median of ABC \triangle ABC drawn to side AC AC .

Therefore, the median of the triangle is BE BE for AC AC .

Answer

BE for AC

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