All terms in triangle calculation

Parts of a Triangle - Examples, Exercises and Solutions
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Terms used in triangle calculations

  • Line

A line is a general term for straight lines (hence its name) that extend from a specific point on the triangle.

  • Height

Height is a line that extends from a specific vertex and reaches perpendicularly to the opposite side, creating a right angle. The height is marked with the letter h (from the word height).

  • Median

The median is also a line extending from a specific vertex to the opposite side, but it reaches exactly the middle of the opposite side and divides it into two equal parts.

  • Angle Bisector

An angle bisector is a line that extends from a specific vertex and actually divides the vertex into two equal angles.

  • Perpendicular Bisector

A perpendicular bisector is a line that extends from the middle of a side perpendicular to it.

  • Midsegment

A midsegment is a line that connects the midpoints of two sides and is parallel to the third side, with its length being half of it.

  • Opposite Side

An opposite side is the side that is located opposite to a specific vertex and does not pass through it.

Diagram of a triangle ABC illustrating key geometric concepts: height (H) in green, median in blue, angle bisector in red, perpendicular bisector from CB in orange, midsegment in purple, and the side opposite to vertex A highlighted in orange. Labels are color-coded for clarity.

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einstein

Is DE side in one of the triangles?
AAABBBCCCDDDEEE

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Terms used in triangle calculations

Line

A ray is a general name for straight lines (hence its name) that extend from a specific point on the triangle.

Height

Height is a line that extends from a specific vertex and reaches perpendicularly to the opposite side, creating a right angle. Height is marked with the letter h (from the word height in English).

Median

The median is also a line that extends from a specific vertex to the opposite side, but it reaches exactly the middle of the opposite side and divides it into two equal parts.

Angle Bisector

An angle bisector is a line that extends from a specific vertex and divides the angle into two equal angles.

Perpendicular Bisector

A median perpendicular is a line that extends from the middle of a side perpendicular to it.

Midsegment

A midsegment is a line segment that connects the midpoints of two sides and is parallel to the third side, and its length is half of it.

Opposite Side

An opposite side is the side located across from a specific vertex and does not pass through it.

Diagram of a triangle ABC illustrating key geometric concepts: height (H) in green, median in blue, angle bisector in red, perpendicular bisector from CB in orange, midsegment in purple, and the side opposite to vertex A highlighted in orange. Labels are color-coded for clarity.

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Test your knowledge
Question 1

The triangle ABC is shown below.

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

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Question 2

Look at triangle ABC below.

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

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Question 3

Look at triangle ABC below.

Which is the median?

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Examples with solutions for Parts of a Triangle

Exercise #1

Given the following triangle:

Write down the height of the triangle ABC.

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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 #2

Which of the following is the height in triangle ABC?

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Video Solution

Step-by-Step Solution

Let's remember the definition of height of a triangle:

A height is a straight line that descends from the vertex of a triangle and forms a 90-degree angle with the opposite side.

The sides that form a 90-degree angle are sides AB and BC. Therefore, the height is AB.

Answer

AB

Exercise #3

ABC is an isosceles triangle.

AD is the median.

What is the size of angle ∢ADC ∢\text{ADC} ?

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Video Solution

Step-by-Step Solution

In an isosceles triangle, the median to the base is also the height to the base.

That is, side AD forms a 90° angle with side BC.

That is, two right triangles are created.

Therefore, angle ADC is equal to 90 degrees.

Answer

90

Exercise #4

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

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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 #5

Can a triangle have two right angles?

Video Solution

Step-by-Step Solution

The sum of angles in a triangle is 180 degrees. Since two angles of 90 degrees equal 180, a triangle can never have two right angles.

Answer

No

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