Question

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) left parenthesis negative 1 comma 1 right parenthesis

Answer 1

To prove that the three medians of a triangle meet at a single point known as the centroid, we can use the properties of medians and their intersection.

### Definitions and Set-Up:
1. A **median** of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
2. In triangle \( ABC \), the medians are \( AD \), \( BE \), and \( CF \), where \( D \), \( E \), and \( F \) are the midpoints of sides \( BC \), \( AC \), and \( AB \) respectively.
3. The point where the three medians intersect is called the **centroid** \( G \).

### Proof:
Let's place triangle \( ABC \) in the coordinate plane for convenience. Let:
- \( A(x_1, y_1) \)
- \( B(x_2, y_2) \)
- \( C(x_3, y_3) \)

#### Finding the Midpoints:
1. The midpoint \( D \) of segment \( BC \) is:
\[
D\left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right)
\]
2. The midpoint \( E \) of segment \( AC \) is:
\[
E\left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right)
\]
3. The midpoint \( F \) of segment \( AB \) is:
\[
F\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

#### Finding the Equations of the Medians:
- The median \( AD \):
To find the equation of median \( AD \), we can use the parametric form of the line or the slope-intercept form. However, we can also find coordinates of \( G \) directly using the section formula.

- Finding the centroid \( G \):
The centroid \( G \) divides each median in the ratio \( 2:1 \), meaning:
\[
G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3}
\]
Therefore the centroid \( G \) has coordinates \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \).

### Conclusion:
Since we have shown that the medians of triangle \( ABC \) intersect at the point \( G \), which is defined by the averages of the vertices’ coordinates, we conclude that all three medians intersect at a single point, the centroid.

### Finding the Centroid in Given Options:
The options provided for the centroid are:
- (−12,12)
- \(\left(-\frac{1}{2}, \frac{1}{2}\right)\)
- (−13,13)
- \(\left(-\frac{1}{3}, \frac{1}{3}\right)\)
- (−3,1)
- (−1,1)

However, without specific coordinates of the vertices \( A \), \( B \), and \( C \), we cannot determine which of these is the correct centroid. We can only assert that the centroid \( G \) is located at \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \).

To select the correct choice from the available options, we need the coordinates of triangle \( ABC \). If you provide those coordinates, I could assist you further in identifying the centroid from the given list.