Class 10 Mathematics – Triangle and Centroid

1. Triangle Basics

• A triangle is a three-sided polygon.

• It has three vertices (corners), three sides, and three angles.

Types of Triangles (by sides)

• Equilateral: All sides equal

• Isosceles: Two sides equal

• Scalene: All sides different

Types of Triangles (by angles)

• Acute: All angles < 90°

• Right: One angle = 90°

• Obtuse: One angle > 90°

2. Medians of a Triangle

• A median is a line joining a vertex of the triangle to the midpoint of the opposite side.

• Every triangle has three medians.

• Properties of medians:

  • They intersect at a single point called the centroid.

  • Divide the triangle into two smaller triangles of equal area.

3. Centroid of a Triangle

• The centroid is the point where all three medians intersect.

• It is considered the center of gravity or balancing point of the triangle.

• Properties of the centroid:

  • Always lies inside the triangle, regardless of triangle type.

  • Divides each median into two segments, with the part closer to the vertex being twice as long as the part closer to the midpoint.

  • Helps in determining the center point for balance, symmetry, and design.

4. Importance of the Centroid

• Used in engineering and architecture to find the center of mass.

• Helps in geometrical constructions and designs.

• Important in coordinate geometry for finding the average position of the vertices.

5. Steps to Identify the Centroid Conceptually

1. Find the midpoint of each side of the triangle.

2. Draw medians from each vertex to the midpoint of the opposite side.

3. The point where all three medians meet is the centroid.

4. Observe that the centroid balances the triangle if it were a physical object.

6. Key Points to Remember

• Centroid = intersection of three medians

• Lies inside the triangle

• Divides each median in 2:1 ratio (vertex side : midpoint side)

• Acts as the center of gravity of the triangle