Parts of a Triangle - Examples, Exercises and Solutions

Understanding Parts of a Triangle

Complete explanation with examples

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.

Detailed explanation

Practice Parts of a Triangle

Test your knowledge with 36 quizzes

Which of the following is the height in triangle ABC?

AAABBBCCCDDD

Examples with solutions for Parts of a Triangle

Step-by-step solutions included
Exercise #1

Look at the triangle ABC below.

AD=12AB AD=\frac{1}{2}AB

BE=12EC BE=\frac{1}{2}EC

What is the median in the triangle?

AAABBBCCCEEEDDD

Step-by-Step Solution

A median in a triangle is a line segment connecting a vertex to the midpoint of the opposite side. Here, we need to find such a segment in triangle ABC \triangle ABC .

Let's analyze the given conditions:

  • AD=12AB AD = \frac{1}{2}AB : Point D D is the midpoint of AB AB .
  • BE=12EC BE = \frac{1}{2}EC : Point E E is the midpoint of EC EC .

Given that D D is the midpoint of AB AB , if we consider the line segment DC DC , it starts from vertex D D and ends at C C , passing through the midpoint of AB AB (which is D D ), fulfilling the condition for a median.

Therefore, the line segment DC DC is the median from vertex A A to side BC BC .

In summary, the correct answer is the segment DC DC .

Answer:

DC

Exercise #2

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

Exercise #3

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCEEEDDD

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

Video Solution
Exercise #4

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 #5

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

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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

Video Solution

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