Question

there's a triangle with 3 points, A(-5,1), B(1,7), and C(1,-5)

Prove that when the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid?

(-1/2, 1/2)
(-1/3, 1/3)
(-1,1)
(-3,1)

Answer 1

To prove that the three medians of triangle ABC meet at a single point, we need to find the centroid of the triangle formed by the points A, B, and C.

The centroid (G) of a triangle with vertices at coordinates A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) can be calculated using the formula:

\[
G\left(x_G, y_G\right) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
\]

In our case, we have:

- A(-5, 1)
- B(1, 7)
- C(1, -5)

Let's calculate the coordinates of the centroid.

1. **x-coordinate of the centroid**:
\[
x_G = \frac{-5 + 1 + 1}{3} = \frac{-5 + 2}{3} = \frac{-3}{3} = -1
\]

2. **y-coordinate of the centroid**:
\[
y_G = \frac{1 + 7 - 5}{3} = \frac{1 + 7 - 5}{3} = \frac{3}{3} = 1
\]

Thus, the centroid G of triangle ABC is:

\[
G(-1, 1)
\]

Now, let's check the options provided to find which one matches the calculated centroid:

- (-1/2, 1/2)
- (-1/3, 1/3)
- (-1, 1) (This matches)
- (-3, 1)

The centroid is **(-1, 1)**. Hence, we have proved that the three medians intersect at a single point, which is the centroid.

The answer is \((-1, 1)\).