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 (−3,1) left parenthesis negative 3 comma 1 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 (−13,13)

Answer 1

To prove that the three medians of a triangle intersect at a single point, we need to show that the medians are concurrent. The point where the three medians meet is called the centroid of the triangle.

Proof:

1. Definition of a Median: A median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side.

2. Let Triangle Vertices: Consider triangle \( ABC \) with vertices at coordinates \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).

3. Find Midpoints:

   • The midpoint \( D \) of side \( BC \) is given by: \[ D\left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) \]

   • The midpoint \( E \) of side \( AC \) is: \[ E\left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \]

   • The midpoint \( F \) of side \( AB \) is: \[ F\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

4. Equations of Medians:

   • The median from vertex \( A \) to midpoint \( D \) can be represented parametrically or by vector equations. The key step is to determine where the medians intersect.

5. Finding the Centroid: The centroid \( G \) of triangle \( ABC \) is calculated as the average of the vertices' coordinates: \[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

6. Intersection of the Medians: By showing the intersection of the medians happens at the coordinates of the centroid \( G \), we demonstrate that the medians are concurrent.

7. Conclusion: Every triangle's medians intersect at a single point called the centroid. The point \( G \) divides each median in a ratio of 2:1.

Identifying Options for Centroid Location:

To find the centroid explicitly, you would need the specific coordinates of vertices \( A \), \( B \), and \( C \). However, from the answer choices provided:

• (-3, 1)
• \((- \frac{1}{2}, \frac{1}{2})\)
• (-1, 1)
• (-13, 13)

You can choose the correct option based on the given triangle's coordinates once they are provided. If you have specific vertex coordinates, compute the centroid using the centroid formula provided and compare it with the choices to find the correct point.

Answer 2

The coordinates are, A is, (-5,1). B is, (1,7). C is, (1,-5)

Answer 3

To find the centroid of triangle \( ABC \) with vertices \( A(-5, 1) \), \( B(1, 7) \), and \( C(1, -5) \), we can use the centroid formula:

\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Step 1: Substitute the coordinates

The coordinates for the vertices are:

• \( A(x_1, y_1) = (-5, 1) \)
• \( B(x_2, y_2) = (1, 7) \)
• \( C(x_3, y_3) = (1, -5) \)

Step 2: Compute the centroid coordinates

1. Calculate the x-coordinate of the centroid: \[ x_G = \frac{-5 + 1 + 1}{3} = \frac{-3}{3} = -1 \]

2. Calculate the y-coordinate of the centroid: \[ y_G = \frac{1 + 7 - 5}{3} = \frac{3}{3} = 1 \]

Conclusion: The Centroid Coordinates

Therefore, the centroid \( G \) of triangle \( ABC \) is: \[ G(-1, 1) \]

Answer

The centroid \( G \) is at the coordinates \( (-1, 1) \). So the correct option from your choices is: \[ (-1, 1) \]