Use the image to answer the question.

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

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

(−3,1)
left parenthesis negative 3 comma 1 right parenthesis

(−1,1)

Answer 1

To find the centroid of a triangle given its vertices, we can use the formula for the centroid $G$ of a triangle with vertices $A(x_1,y_1)$, $B(x_2,y_2)$, and $C(x_3,y_3)$:

$$G(x,y)=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$$

Without the specific coordinates of vertices $A$, $B$, and $C$ provided in your description, I'll use generic coordinates $A(x_1,y_1)$, $B(x_2,y_2)$, and $C(x_3,y_3)$ to show that the medians meet at a single point: the centroid.

Calculating $G$:

$$x_G=\frac{x_1+x_2+x_3}{3}$$ $$y_G=\frac{y_1+y_2+y_3}{3}$$

The medians connect each vertex with the midpoint of the opposite side, and by definition, they all intersect at the centroid, proving that the three medians meet at a single point.

Now to determine the centroid's coordinates, you'd substitute in the specific values for $x_1,y_1,x_2,y_2,x_3,$ and $y_3$ based on the triangle's vertices.

Since the specific coordinates were not provided in your question, I can't compute an explicit coordinate for the centroid. However, based on common choices given, you would compute the centroid’s coordinates to match one of the provided responses.

If you provide the coordinates of the triangle's vertices, I can help calculate the exact location of the centroid. Otherwise, without that information, I would only be able to present the method as shown. Please share the coordinates of the vertices if available.