Practical geometry: centroid definition

• What is a centroid: Centroid of a triangle is the center point of a triangle. 
• A triangle is a three-sided figure with three interior angles, three sides and and three vertices.  
• Based on the sides and angles, a triangle can be classified into different types:
  • Equilateral triangle
  • Scalene triangle
  • Acute-angled triangle
  • Obtuse-angled triangle
  • Isosceles triangle
  • Right-angled triangle
• Centroid definition: Centroid is the point of intersection of all the three medians. 
• The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. 
• The centroid of the triangle separates the median in the ratio of 2:1. 
• Orthocenter is the location where three altitudes of a triangle meet. 
• Circumcenter is the point of meeting of three perpendicular bisectors. 
• Perpendicular bisector is the line drawn perpendicularly from the midpoint of a triangle. 
• Incenter is the point of intersection of angle bisectors. 
• An orthocenter can be outside of a triangle. 
• An incenter is always inside the triangle. 
• A centroid is also always inside the triangle. 
• A centroid exists for other shapes, like squares. 

Questions to practice:

Draw medians and verify that they meet at a concurrent point within the triangle. Also show that medians divide each other in the ratio 2:1. 

Draw angle bisectors and verify concurrency. 

Questions for review:

• What is the centroid of a triangle?
• Draw angle bisectors and verify concurrency. 
• What is a median? Draw medians of a given triangle and verify concurrency. 
• What is a perpendicular bisector? Draw perpendicular bisectors and verify concurrency. 
• What is the ratio of medians dividing each other? 

Further thinking: 

• Is there a centroid for a square?
• Is it true that centroid is the point of concurrency?
• Can a centroid be outside of a triangle?
• Can an orthocenter be outside of a triangle?
• Can you find a centroid for a right triangle? 
• Do all types of triangles display a 2:1 ratio of medians? 
• If you make a mistake in real measurements in constructing a triangle, the medians will meet at a different point. Is that true? 
• What is the theorem of centroid using coordinates? 
• What is the relationship between orthocenter, centroid and circumcenter?
• What is the difference between incenter and centroid?
• What is the difference between the orthocenter and centroid of a triangle?

Additional practice questions on constructing other geometrical forms:

1. Construct a rectangle whose adjacent sides are 2.5 cm and 5 cm respectively. Construct a square having area equal to the given rectangle.
2. Construct a square equal in area to a rectangle whose adjacent sides are 4.5 cm and 2.2 cm respectively. Measure the sides of the square and find its area and compare with the area of the rectangle.
3. In Q.2 above verify by measurement that the perimeter of the square is less than that of the rectangle.
4. Construct a square equal in area to the sum of two squares having sides 3 cm and 4 cm respectively. (Source: Math class 9 book, Punjab Text Book Board)

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