0Class 10 Mathematics – Internal & External Division with Centroid
1. Line Segment Division
A line segment connects two points in a coordinate plane. Sometimes we need to find a specific point on this line that divides it in a certain ratio.
a) Internal Division
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A point divides a line segment internally if it lies between the two endpoints.
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Example: A point P lies on the line joining A and B, somewhere between A and B.
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Applications:
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Finding a point halfway between two points (midpoint).
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Dividing a property, road, or field in a specific proportional ratio.
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b) External Division
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A point divides a line segment externally if it lies outside the segment, beyond one of the endpoints.
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Example: A point Q is beyond A or B along the same line.
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Applications:
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Extending a line beyond endpoints to create proportional sections.
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Engineering and architecture to locate extended points or supports.
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2. Centroid of a Triangle
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The centroid is the point where all three medians of a triangle intersect.
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Median: A line joining a vertex of the triangle to the midpoint of the opposite side.
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The centroid is considered the center of gravity of the triangle.
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Applications:
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Balancing a triangular object
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Finding the geometrical center in land division, design, or construction
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3. How Internal & External Division Relate to the Centroid
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The centroid divides each median in a 2:1 ratio internally, meaning the section closer to the vertex is twice as long as the part closer to the midpoint of the opposite side.
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This is a special case of internal division in coordinate geometry.
4. Key Points to Remember
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Internal division: Point lies between endpoints.
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External division: Point lies outside endpoints.
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Centroid: Intersection of medians; acts as triangle’s balance point.
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Section formula concepts help in finding coordinates of points dividing a line segment internally or externally.
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Useful in geometry problems, land division, and design applications.
