Find the co-ordinates of the centroid of a triangle whose vertices are A(1, 4), B(7, 8) and C(10, 12).
Centroid Coordinates Calculation for Triangle Vertices
The problem asks us to find the coordinates of the centroid of a triangle given its vertices A, B, and C.
Understanding the Centroid
The centroid is the point where the three medians of a triangle intersect. A median connects a vertex to the midpoint of the opposite side. The centroid is also known as the center of gravity or the geometric center of the triangle.
Centroid Formula
For a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the coordinates of the centroid (G) are calculated using the following formula:
$$ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) $$Given Vertices
- Vertex A: $(x_1, y_1) = (1, 4)$
- Vertex B: $(x_2, y_2) = (7, 8)$
- Vertex C: $(x_3, y_3) = (10, 12)$
Step-by-Step Calculation
Calculating the x-coordinate of the Centroid:
We sum the x-coordinates of the vertices and divide by 3:
$$ \text{Centroid x} = \frac{x_1 + x_2 + x_3}{3} $$ $$ \text{Centroid x} = \frac{1 + 7 + 10}{3} $$ $$ \text{Centroid x} = \frac{18}{3} $$ $$ \text{Centroid x} = 6 $$Calculating the y-coordinate of the Centroid:
We sum the y-coordinates of the vertices and divide by 3:
$$ \text{Centroid y} = \frac{y_1 + y_2 + y_3}{3} $$ $$ \text{Centroid y} = \frac{4 + 8 + 12}{3} $$ $$ \text{Centroid y} = \frac{24}{3} $$ $$ \text{Centroid y} = 8 $$Result
The calculated coordinates of the centroid are $(6, 8)$.
Matching with Options
Comparing our result with the given options:
- Option 1: (7,8)
- Option 2: (6, 8)
- Option 3: (6, 9)
- Option 4: (7, 9)
The calculated coordinates $(6, 8)$ match Option 2.
Important Questions from Coordinate Geometry
- If x² + y² - 16x + 38y + 425 = 0, then the value of x² + y² is:
- If x² + y² - 12x + 18y + 117 = 0, then the value of x² + y² is:
- A geological survey team has established three research stations forming a triangular monitoring zone. The stations are located at coordinates P(2,5), Q(8,1), and R(4,9) on a coordinate grid where each unit represents 1 kilometer. What is the area of the triangular monitoring zone?
- \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\) and \(D(6, 14)\) are the vertices of a parallelogram, taken in order, find the value of \(a\).
- What would be the area of the triangle with \(A(8, 8)\), \(B(22, 8)\) and \(C(16, -10)\) as vertices?
