For the next three (3) items that follow:
What is the centroid of the triangle ABC?
Finding the Centroid of a Triangle Using Vertices
The question asks us to find the centroid of triangle ABC with given vertices A(-2, 3), B(2, 1), and C(1, 2).
The centroid of a triangle is a fundamental point in geometry. It is the point where the three medians of the triangle intersect. A median connects a vertex to the midpoint of the opposite side. The centroid represents the geometric center of the triangle.
Given the coordinates of the vertices of a triangle, say \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\), the coordinates of the centroid \(G(x, y)\) can be found using the following formula:
Centroid \(G(x, y) = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\)
Now, let's apply this formula to the given vertices of triangle ABC:
- Vertex A: \((x_1, y_1) = (-2, 3)\)
- Vertex B: \((x_2, y_2) = (2, 1)\)
- Vertex C: \((x_3, y_3) = (1, 2)\)
First, we calculate the x-coordinate of the centroid:
\(x = \frac{x_1 + x_2 + x_3}{3}\)
\(x = \frac{-2 + 2 + 1}{3}\)
\(x = \frac{1}{3}\)
Next, we calculate the y-coordinate of the centroid:
\(y = \frac{y_1 + y_2 + y_3}{3}\)
\(y = \frac{3 + 1 + 2}{3}\)
\(y = \frac{6}{3}\)
\(y = 2\)
So, the coordinates of the centroid of triangle ABC are \(\left( \frac{1}{3}, 2 \right)\).
Let's compare this result with the given options to identify the correct centroid coordinates.
| Option | Centroid Coordinates |
|---|---|
| 1 | \(\left( {\frac{1}{3},1} \right)\) |
| 2 | \(\left( {\frac{1}{3},2} \right)\) |
| 3 | \(\left( {1,\frac{2}{3}} \right)\) |
| 4 | \(\left( {\frac{1}{2},{\rm{\;}}3} \right)\) |
Our calculated centroid is \(\left( \frac{1}{3}, 2 \right)\), which matches the coordinates in Option 2.
Revision Table: Key Concepts for Centroid Calculation
| Concept | Description | Formula (for vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\)) |
|---|---|---|
| Centroid | Intersection point of the triangle's medians. Represents the geometric center. | \(G\left( {\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}} \right)\) |
| Median | A line segment joining a vertex to the midpoint of the opposite side. | N/A (Line segment, not a single point coordinate formula) |
| Midpoint | The point exactly halfway between two given points \((x_a, y_a)\) and \((x_b, y_b)\). | \(M\left( {\frac{x_a + x_b}{2}, \frac{y_a + y_b}{2}} \right)\) |
Additional Information on Triangle Centroid and Geometry
The centroid is one of the four main centers of a triangle, along with the incenter, circumcenter, and orthocenter. It has several interesting properties:
- The centroid always lies inside the triangle.
- The centroid divides each median in a 2:1 ratio, with the longer segment being between the vertex and the centroid. For example, if M is the midpoint of BC, the centroid G divides AM such that AG : GM = 2 : 1.
- If the triangle were made of a thin, uniform material, the centroid would be its center of mass or balance point.
- The centroid's coordinates are the average of the coordinates of the vertices. This property makes the formula very intuitive and easy to remember.
Understanding how to find the centroid is important in coordinate geometry and has applications in physics (center of mass) and engineering.
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