Question

Medians of a Triangle Quick Check 5 of 55 of 5 Items 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 (13,0) left parenthesis Start Fraction 1 over 3 End Fraction comma 0 right parenthesis (16,3) left parenthesis Start Fraction 1 over 6 End Fraction comma 3 right parenthesis (0,13) left parenthesis 0 comma Start Fraction 1 over 3 End Fraction right parenthesis (13,1) left parenthesis Start Fraction 1 over 3 End Fraction comma 1 right parenthesis Skip to navigation page 8 of 8

Answer 1

To find the centroid of triangle ABC, we use the formula for the centroid (G) of a triangle given its vertices 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 of triangle ABC:
- A(−3,−3)
- B(0,6)
- C(4,−3)

We can plug these coordinates into the formula:

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

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

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

From the provided options, the corresponding centroid is:
\[
\left( \frac{1}{3}, 0 \right)
\]

Therefore, the medians of triangle ABC meet at the point \( \left( \frac{1}{3}, 0 \right) \).