Question

A coordinate plane's axes range from negative 7 to 7, both by 1-unit increments. A triangle made up of vertices upper A, upper B, and upper C is plotted. The coordinates of the vertices are given.
Prove that when the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid?

(1 point)
Responses

(−12,12)
left parenthesis negative Start Fraction 1 over 2 End Fraction comma Start Fraction 1 over 2 End Fraction right parenthesis

(−1,1)
left parenthesis negative 1 comma 1 right parenthesis

(−13,13)
left parenthesis negative Start Fraction 1 over 3 End Fraction comma Start Fraction 1 over 3 End Fraction right parenthesis

(−3,1)

Answer 1

To find the centroid of triangle ABC, we can use the formula for the centroid (G) of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃):

\[
G\left(\frac{x₁ + x₂ + x₃}{3}, \frac{y₁ + y₂ + y₃}{3}\right)
\]

The centroid is the point where the three medians of the triangle intersect, and it is always located at this calculated point.

Let's denote the coordinates of the triangle's vertices:
- Vertex A: \( (x_1, y_1) \)
- Vertex B: \( (x_2, y_2) \)
- Vertex C: \( (x_3, y_3) \)

We will calculate the centroid given the coordinates of these points. However, since the coordinates of the vertices (A, B, and C) were not provided in your inquiry, I will describe the steps to determine the centroid once you know the coordinates:

1. **Add the x-coordinates of the vertices**:
\[
x_{sum} = x_1 + x_2 + x_3
\]

2. **Add the y-coordinates of the vertices**:
\[
y_{sum} = y_1 + y_2 + y_3
\]

3. **Calculate the x-coordinate of the centroid**:
\[
G_x = \frac{x_{sum}}{3}
\]

4. **Calculate the y-coordinate of the centroid**:
\[
G_y = \frac{y_{sum}}{3}
\]

The final result will give you the coordinates of the centroid.

Let's also consider the options provided in your query. The centroid can only be one of the given options. Once the calculations are completed with the coordinates, you can compare your result to each option. If needed, based on typical coordinates in a reasonable range given the problem constraints, the answer could be any of the options provided.

If you provide the coordinates of points A, B, and C, I can help you calculate the exact centroid.