Centroid
The centroid of a triangle is the point where the three medians of the triangle intersect. The median of a triangle is a line segment from a vertex to the midpoint of the opposite side.
Finding the Centroid
To find the centroid of a triangle, you can use the following steps:
- Find the midpoint of each side of the triangle.
- Draw a line from each vertex to the midpoint of the opposite side. These lines are the medians of the triangle.
- The point where the three medians intersect is the centroid of the triangle.
Properties of the Centroid
Some important properties of the centroid include:
- The centroid divides each median into two segments, with the segment closer to the midpoint of the opposite side being twice as long as the other segment.
- The centroid is the center of mass of the triangle, meaning it is the point where the triangle would balance if it were cut out of a sheet of uniform material.
Practice Problems
1. Find the centroid of a triangle with vertices at (1, 2), (3, 4), and (5, 6).
2. In triangle ABC, the length of median AD is 6 cm. If D is the centroid of the triangle, find the length of BC.
Additional Resources
For more practice problems and in-depth explanations, check out the following resources:
.◂Math Worksheets and Study Guides Seventh Grade. Plane Figures: Closed Figure Relationships
Study Guide
Plane Figures: Closed Figure Relationships 
Worksheet/Answer key
Plane Figures: Closed Figure Relationships Worksheet/Answer key
Plane Figures: Closed Figure Relationships Worksheet/Answer key
Plane Figures: Closed Figure Relationships
Worksheet/Answer keyWorksheet/Answer keyWorksheet/Answer keyThe resources above cover the following skills:
Geometry (NCTM)
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
Connections to the Grade 7 Focal Points (NCTM)
Data Analysis: Students use proportions to make estimates relating to a population on the basis of a sample. They apply percentages to make and interpret histograms and circle graphs.