Question

Triangle ABC is drawn on a coordinate plane with vertices A(−3,−3), B(0,6), and C(4,−3) and medians indicating the midpoint of each of the lines AB, BC, and CA. Prove the medians meet at a single point by finding the centroid.(1 point)
Responses

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

(0,13)
left parenthesis 0 comma Start Fraction 1 over 3 End Fraction right parenthesis

(13,0)
left parenthesis Start Fraction 1 over 3 End Fraction comma 0 right parenthesis

(16,3)

Answer 1

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

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

Given the vertices:
- A(-3, -3)
- B(0, 6)
- C(4, -3)

We can substitute these coordinates into the centroid formula:

1. Calculate the x-coordinate of the centroid:
\[
G_x = \frac{-3 + 0 + 4}{3} = \frac{1}{3}
\]

2. Calculate the y-coordinate of the centroid:
\[
G_y = \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0
\]

Putting it all together, the coordinates of the centroid G are:
\[
G\left(\frac{1}{3}, 0\right)
\]

Therefore, the centroid is located at:

\[
\left(\frac{1}{3}, 0\right)
\]

From the provided responses, the correct answer is:

**(13,0)** (Note: this is likely a typo or misrepresentation. Based on the calculation, the centroid is \(\left(\frac{1}{3}, 0\right)\).