Points to Remember:
- Centroid coordinates are the average of the x-coordinates and y-coordinates of the vertices.
- The formula for the centroid (G) of a triangle with vertices A(xâ, yâ), B(xâ, yâ), and C(xâ, yâ) is: G = ((xâ + xâ + xâ)/3, (yâ + yâ + yâ)/3)
Introduction:
This question requires a factual and analytical approach to determine the coordinates of the third vertex of a triangle given the coordinates of the centroid and two vertices. The centroid of a triangle is the point where the three medians of the triangle intersect. A median is a line segment from a vertex to the midpoint of the opposite side. The centroid’s coordinates are the average of the x-coordinates and the average of the y-coordinates of the three vertices.
Body:
1. Understanding the Centroid Formula:
The centroid (G) of a triangle with vertices A(xâ, yâ), B(xâ, yâ), and C(xâ, yâ) is given by:
G = ((xâ + xâ + xâ)/3, (yâ + yâ + yâ)/3)
2. Applying the Formula:
We are given:
- Centroid G = (3, 2)
- Vertex A = (1, 1)
- Vertex B = (2, 5)
- Let Vertex C = (x, y)
Substituting these values into the centroid formula, we get two equations:
- (1 + 2 + x)/3 = 3 => 3 + x = 9 => x = 6
- (1 + 5 + y)/3 = 2 => 6 + y = 6 => y = 0
Therefore, the coordinates of the third vertex C are (6, 0).
3. Verification:
Let’s verify our result by calculating the centroid using the coordinates of all three vertices:
Centroid = ((1 + 2 + 6)/3, (1 + 5 + 0)/3) = (9/3, 6/3) = (3, 2)
This matches the given centroid coordinates, confirming our calculation.
Conclusion:
By applying the formula for the centroid of a triangle and using the given coordinates of the centroid and two vertices, we have successfully determined the coordinates of the third vertex to be (6, 0). This problem demonstrates a fundamental concept in coordinate geometry and highlights the application of simple algebraic manipulation to solve geometric problems. The accuracy of the solution is verified by recalculating the centroid using the derived coordinates of the third vertex. This approach emphasizes the importance of precise calculations and the application of established geometric principles. Further exploration of centroid properties could involve investigating their applications in areas like computer graphics and engineering design.