Question

Use the image to answer the 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

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

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

(−12,12)
left parenthesis negative Start Fraction 1 over 2 End Fraction comma Start Fraction 1 over 2 End Fraction right parenthesis

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

The three medians of a triangle concur at the centroid, which has coordinates equal to the average of the vertices' coordinates.

Proof (coordinate form). Let the vertices be A(x1,y1), B(x2,y2), C(x3,y3). The midpoint of BC is ((x2+x3)/2,(y2+y3)/2). The median from A consists of points
A + t( midpoint(BC) − A ) = ( x1 + t( (x2+x3)/2 − x1 ), y1 + t( (y2+y3)/2 − y1 ) ).
Setting t = 2/3 gives the point
( (x1+x2+x3)/3 , (y1+y2+y3)/3 ).
Doing the same for the medians from B and C produces the same point, so all three medians meet there. That point is the centroid.

Using the triangle's vertex coordinates from the figure and averaging them, the centroid is (−3, 1).

Answer: (−3, 1)