Question

medians of a triangle quick check
triangle abc is drawn on a coordinate plane with vertices ( a(-2,-3) ), ( b(4,1) ), and ( c(-2,2) ) and medians indicating the midpoint of each of the line segments ( overline{ab} ), ( overline{bc} ), and ( overline{ca} ). prove that the medians meet at a single point by finding the centroid (1 point)
( (-1,\frac{1}{3}) )
( (0,1) )
( (0,-\frac{1}{3}) )
( (0,0) )

Explanation:

Step1: Recall the centroid formula

The centroid formula for a triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) is \((\frac{x_1 + x_2+x_3}{3},\frac{y_1 + y_2 + y_3}{3})\)

Step2: Identify the coordinates of vertices

Given \(A(-2,-3)\), \(B(4,1)\), \(C(-2,2)\), so \(x_1=-2\), \(y_1=-3\), \(x_2 = 4\), \(y_2=1\), \(x_3=-2\), \(y_3 = 2\)

Step3: Calculate the \(x\) - coordinate of centroid

$$
LATEXBLOCK0
$$

Step4: Calculate the \(y\) - coordinate of centroid

$$
LATEXBLOCK1
$$

So the centroid is \((-1,\frac{2}{3})\)

Answer:

A. \((-1,\frac{2}{3})\)