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.

Practice Parts of a Triangle

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

Exercise #6

Look at triangle ABC below.

Which is the median?

αααAAABBBCCCDDDEEE

Step-by-Step Solution

To solve this problem, we must identify which line segment in triangle ABC is the median.

First, review the definition: a median in a triangle connects a vertex to the midpoint of the opposite side. Now, in triangle ABC:

  • Point A represents the vertex.
  • Point E lies on line segment AB.
  • Line segment EC needs to be checked to see if it connects vertex E to point C.

From the diagram, it appears that E is indeed the midpoint of side AB. Thus, line segment EC connects vertex C to this midpoint.

This fits the definition of a median, verifying that EC is the median line segment in triangle ABC.

Therefore, the solution to the problem is: EC \text{EC} .

Answer

EC

Exercise #7

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 #8

ABC is a triangle.

What is the median of the triangle?

AAABBBCCCEEEFFFDDD

Step-by-Step Solution

To solve the problem of identifying the median of triangle ABC \triangle ABC , we follow these steps:

  • Step 1: Understand the Definition - A median of a triangle is a line segment that extends from a vertex to the midpoint of the opposite side.
  • Step 2: Identify Potential Medians - Examine segments from each vertex to the opposite side. The diagram labels these connections.
  • Step 3: Confirm the Median - Specifically check the segment EC in the context of the line segment from vertex E E to the side AC AC , and verify it reaches the midpoint of side AC AC .
  • Step 4: Verify Against Options - Given choices allow us to consider which point-to-point connection adheres to our criterion for a median. EC is given as one of the choices.

Observation shows: From point E E (assumed from the label and position) that line extends directly to point C C —a crucial diagonal opposite from considered midpoint indications, suggesting it cuts AC AC evenly, classifying it as a median.

Upon reviewing the given choices, we see that segment EC EC is listed. Confirming that EC EC indeed meets at C C , the midpoint of AC AC , validates that it is a true median.

Therefore, the correct median of ABC \triangle ABC is the segment EC EC .

Answer

EC

Exercise #9

Look at the triangles in the figure.

Which line is the median of triangle ABC?

AAABBBCCCDDDEEEGGGFFF

Step-by-Step Solution

To determine the median of triangle ABC ABC , we need to identify the line that extends from one vertex to the midpoint of the opposite side.

  • Step 1: Review the given line segments in the figure.
  • Step 2: Recall that a median connects a vertex to the midpoint of the opposite side.
  • Step 3: Examine each line in the context of ABC\triangle ABC.

Let's consider each given line:

  • Line AF AF does not appear to connect to the midpoint of any side of the triangle directly.
  • Line DE DE is an internal line and does not serve as a median of the main triangle ABC ABC .
  • Line FE FE is similar to DE DE , serving non-median purposes interior to another structure.
  • Line AG AG starts at vertex A A and extends to point G G , lying on side BC BC . If G G is the midpoint of BC BC , then AG AG qualifies as the median.

Verification: Point G G is positioned directly between points B B and C C along line BC BC , confirming its role as the midpoint.

Thus, the line AG AG is indeed the median of triangle ABC ABC since it fulfills connecting vertex A A and the midpoint of side BC BC .

Therefore, the solution to the problem is AG AG as the median of triangle ABC ABC .

Answer

AG

Exercise #10

What is the median of triangle ABC?

AAABBBDDDCCCEEEFFF

Step-by-Step Solution

To determine the median of triangle ABC, we must identify a segment connecting a vertex of the triangle to the midpoint of the opposite side.

Examining the diagram, point F appears to be located on side AC. Given the configuration, point F divides side AC into two equal segments, which makes F the midpoint of AC.

Therefore, segment CF connects vertex C to the midpoint F of side AC. This characteristic aligns with the definition of a median in a triangle.

Hence, the median of triangle ABC is CF CF .

Answer

CF

Exercise #11

What is the median of triangle ABC.

2X2X2XXXXAAACCCBBBDDDEEE

Step-by-Step Solution

In this problem, we must determine if any of the line segments drawn within triangle ABC represent a median. A median is defined as a line segment extending from a vertex to the midpoint of the opposite side.

Upon examining the geometry of triangle ABC presented in the diagram:

  • The segment extending from A to the base BC and those depicted from B or C should be checked if they connect to a midpoint on the opposite side.
  • To be considered a median, a line from a vertex must bisect the opposite side into two equal lengths.

None of the segments drawn directly bisect the opposite sides they connect to, as evidenced by either lack of midpoint marking or unequal line segment sections along BC, CA, or AB.

Therefore, after careful inspection, there is no median shown in the given diagram.

Answer

There is no median shown.

Exercise #12

Look at the triangle ABC below.

Which of the following lines is the median of the triangle?

AAABBBCCCDDDEEE

Step-by-Step Solution

To solve this problem, we apply the definition of a median in a triangle. A median is a line segment drawn from a vertex to the midpoint of the opposite side. In the diagram of the triangle ABC:

  • Line AD AD originates from vertex A A and is directed towards point D D on side BC BC .
  • We need to check if D D is the midpoint of BC BC .
  • Given that AD AD meets the definition of a median by dividing BC BC into two equal segments, it is indeed the median.

After evaluating the possible choices:

  • Choice 1: AD AD is a line from A A to the midpoint of BC BC .
  • Choice 2: AE AE doesn't bisect any side.
  • Choice 3: EC EC is not a median.
  • Choice 4: AC AC does not connect a vertex to a midpoint of the opposite side.

Therefore, the solution to the problem is that line segment AD is the median of triangle ABC.

Answer

AD

Exercise #13

Look at the triangle ABC below.

Which of the line segments is the median?

AAABBBCCCGGGHHHFFFDDDEEE

Step-by-Step Solution

To identify the median in triangle ABCABC, we will utilize the definition of a median: it is the line segment extending from a vertex of the triangle to the midpoint of the opposite side.

In the diagram, triangle ABCABC is formed with vertices AA, BB, and CC. We need to identify which of the segments is drawn from a vertex and intersects the opposite side at its midpoint.

Examine segment FCFC:

  • FF appears to be a midpoint of side ABAB of the triangle ABCABC.
  • Line segment FCFC originates from vertex CC and extends to FF.

The segment FCFC meets the criteria for a median as it connects vertex CC to the midpoint of ABAB.

Therefore, we conclude that the median of triangle ABCABC is FC.

Answer

FC

Exercise #14

True or false:

AB is a side of the triangle ABC.

AAABBBCCC

Video Solution

Step-by-Step Solution

To solve this problem, let's clarify the role of AB in the context of triangle ABC by analyzing its diagram:

  • Step 1: Identify the vertices of the triangle. According to the diagram, the vertices of the triangle are points labeled A, B, and C.
  • Step 2: Determine the sides of the triangle. In any triangle, the sides are the segments connecting pairs of distinct vertices.
  • Step 3: Identify AB as a line segment connecting vertex A and vertex B, labeled directly in the diagram.

Considering these steps, line segment AB connects vertex A with vertex B, and hence, forms one of the sides of the triangle ABC. Therefore, AB is indeed a side of triangle ABC as shown in the diagram.

The conclusion here is solidly supported by our observation of the given triangle. Thus, the statement that AB is a side of the triangle ABC is True.

Answer

True

Exercise #15

True or false:

AD is a side of triangle ABC.

AAABBBCCC

Video Solution

Step-by-Step Solution

To determine if line segment AD is a side of triangle ABC, we need to agree on the definition of a triangle's side. A triangle consists of three sides, each connecting pairs of its vertices. In triangle ABC, these sides are AB, BC, and CA. Each side is composed of a direct line segment connecting the listed vertices.

In the diagram provided, there is no indication of a point D connected to point A or any other vertex of triangle ABC. To claim AD as a side, D would need to be one of the vertices B or C, or a commonly recognized point forming part of the triangle’s defined structure. The provided figure and description do not support that AD exists within the given triangle framework, as no point D is defined within or connecting any existing vertices.

Therefore, according to the problem's context and based on the definition of the sides of a triangle, AD cannot be considered a side of triangle ABC. It follows that the statement "AD is a side of triangle ABC" should be deemed not true.

Answer

Not true