Prove that when the three medians of the triangle are drawn, they meet at a single point. What is the point of the centroid?

Question

(1 point)
Responses

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

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

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

(−12,12)

Answer 1

To show that the three medians of a triangle meet at a single point, we can utilize the concept of the centroid.

Proof:

1. Definition of a Median: A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. Each triangle has three medians, one from each vertex.

2. Construction: Let us consider a triangle with vertices \( A \), \( B \), and \( C \). The midpoint of side \( BC \) is denoted as \( M \), the midpoint of side \( AC \) as \( N \), and the midpoint of side \( AB \) as \( P \).

3. Coordinates: Assume the coordinates of the vertices are:

   • \( A(a_1, a_2) \)
   • \( B(b_1, b_2) \)
   • \( C(c_1, c_2) \)

   The midpoints are calculated as follows:

   • \( M = \left( \frac{b_1 + c_1}{2}, \frac{b_2 + c_2}{2} \right) \)
   • \( N = \left( \frac{a_1 + c_1}{2}, \frac{a_2 + c_2}{2} \right) \)
   • \( P = \left( \frac{a_1 + b_1}{2}, \frac{a_2 + b_2}{2} \right) \)

4. Finding the Centroid: The centroid \( G \) of the triangle (the point where all three medians intersect) is given by the formula: \[ G = \left( \frac{a_1 + b_1 + c_1}{3}, \frac{a_2 + b_2 + c_2}{3} \right) \]

5. Properties of Medians: Each median divides the triangle into two smaller triangles of equal area. Thus, medians concur at the centroid, \( G \).

Conclusion:

The centroid of the triangle is the common point where all three medians intersect. It also serves as the center of mass for a uniform triangular plate.

Find the coordinates of the Centroid:

Given the provided options don't refer to specific triangles or coordinates, we cannot definitively determine a specific centroid without additional coordinates.

Options Given:

1. \( (-13, 13) \)
2. \( \left(-\frac{1}{3}, \frac{1}{3}\right) \)
3. \( (-1, 1) \)
4. \( (-3, 1) \)
5. \( (-12, 12) \)

Without specific coordinates for vertices \( A \), \( B \), and \( C \), we cannot identify which option corresponds to the centroid. Each provided coordinate pair could represent a centroid depending on the triangle's vertices.

In general, the centroid's coordinates would depend on the specific vertices of the triangle, and none of these options are inherently tied to a universally valid triangle. More information or specifics about the triangle's vertices would be necessary to confidently select which option is the centroid of a particular triangle.