Triangle Centroid Practice Problems & Median Intersection

Master triangle centroid and median intersection with step-by-step practice problems. Learn the 2:1 ratio theorem and solve geometry problems confidently.

📚Practice Triangle Centroid and Median Properties
  • Find the centroid using median intersection points in triangles
  • Apply the 2:1 ratio theorem to solve for median segments
  • Calculate triangle perimeters using median properties and equal side divisions
  • Identify medians and prove centroid locations in geometric problems
  • Solve for unknown lengths using centroid division properties
  • Work with coordinate geometry to find triangle centers and medians

Understanding Parts of a Triangle

Complete explanation with examples

The center of the triangle

  1. All three medians in a triangle intersect at a single point called the centroid -
    If two medians intersect at a point inside the triangle, the third median must pass through it as well.
  2. The intersection point of the medians - the centroid - divides each median in a ratio of 2:12:1 where the larger part of the median is closer to the vertex.

Diagram of a rectangle labeled ABCD with a marked midpoint M at the intersection of its diagonals. The rectangle is black with white and orange highlights, showcasing symmetry and geometry properties.

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?

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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?

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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

Frequently Asked Questions

What is the centroid of a triangle and how do you find it?

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The centroid is the intersection point where all three medians of a triangle meet. To find it, draw medians from each vertex to the midpoint of the opposite side - they will all intersect at one point, which is the centroid.

How does the centroid divide each median in a triangle?

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The centroid divides each median in a 2:1 ratio, where the longer segment is always closer to the vertex. If a median has length 9, the centroid splits it into segments of length 6 (vertex side) and 3 (base side).

What are the key properties of triangle medians I need to remember?

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Key median properties include: 1) A median connects a vertex to the midpoint of the opposite side, 2) All three medians intersect at the centroid, 3) The centroid divides each median in a 2:1 ratio, 4) Medians divide the triangle into six smaller triangles of equal area.

How do you solve triangle centroid problems step by step?

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Follow these steps: 1) Identify which segments are medians, 2) Locate the centroid where medians intersect, 3) Apply the 2:1 ratio to find unknown segments, 4) Use the fact that medians bisect opposite sides to find equal segments, 5) Calculate perimeters or areas as needed.

Why do all three medians of a triangle always meet at one point?

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This is a fundamental theorem in geometry. If two medians intersect at a point inside the triangle, the third median must pass through that same point. This intersection point is called the centroid and represents the triangle's center of mass.

What's the difference between centroid, circumcenter, and incenter of a triangle?

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The centroid is where medians meet and divides them 2:1. The circumcenter is where perpendicular bisectors meet and is equidistant from vertices. The incenter is where angle bisectors meet and is equidistant from all sides.

How do you find the perimeter of a triangle using median properties?

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Use the fact that medians bisect opposite sides to create equal segments. If you know some side lengths from median endpoints, you can determine the full sides since medians create midpoints. Add all three complete side lengths for the perimeter.

What are common mistakes students make with triangle centroid problems?

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Common errors include: confusing the 2:1 ratio direction (remember the longer part is toward the vertex), not recognizing that medians create equal segments on opposite sides, and mixing up different triangle centers like centroid vs circumcenter.

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