Question

One way to prove that the medians of a triangle all meet at one point is by using arbitrary coordinates (0, 0) , (a, 0) , and (b, c) to represent the three vertices of the triangle. Drag and drop the missing pieces of each step to complete the proof that all three medians intersect at point P.

(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Step 1: Draw the three medians of the triangle. Draw each median so that it starts at a Response area and ends at the Response area of the opposite side.

Step 2: Find the coordinates of each midpoint by taking the sum of the x-coordinates and dividing by 2 and taking the sum of the y-coordinates and dividing by 2.

The midpoint between (0, 0) and (b, c) is (b2, c2). The midpoint of (0, 0) and (a, 0) is (a2, 0). The midpoint of (a, 0) and (b, c) is (a+b2, c2).

Step 3: Find the coordinates of P, the point where the medians appear to intersect, on each median. Use the formula Response area.

Step 4: Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint (a+b2, c2).

13(0, 0)+23(a+b2, c2)=(0, 0) + (a+b3, c3)=(a+b3, c3)
Find the coordinates of P on the median that starts at vertex Response area and ends at midpoint (b2, c2).

13(a, 0)+23(b2, c2)=(a3, 0)+(b3,c3)=(a+b3, c3)
Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint (a2, 0).

13(b, c)+23(a2, 0)=(b3,c3)+(a3,0)=(a+b3,c3)
Step 5: The coordinates of P on each median are Response area, which proves that the three medians of this generic triangle all intersect at the same point.

Answer 1

Triangle ABC is drawn on a coordinate plane with vertices A(−2,−3) , B(4, 0) , and C(−2, 2) and medians indicating the midpoint of each of the line segments AB¯¯¯¯¯¯¯¯ , BC¯¯¯¯¯¯¯¯ , and CA¯¯¯¯¯¯¯¯ . Prove that the medians meet at a single point by finding the centroid. (1 point)
Responses

(0, 53)
open paren 0 comma 5 thirds close paren

(−1, 53)
open paren negative 1 comma 5 thirds close paren

(0, 0)
open paren 0 comma 0 close paren

(0,−13)

Answer 2

Triangle ABC is drawn on a coordinate plane with vertices A(−3,−3) , B(0,6) , and C(4,−3) and medians indicating the midpoint of each of the lines AB , BC , and CA . Prove the medians meet at a single point by finding the centroid.(1 point)
Responses

(0,13)
left parenthesis 0 comma Start Fraction 1 over 3 End Fraction right parenthesis

(16,3)
left parenthesis Start Fraction 1 over 6 End Fraction comma 3 right parenthesis

(13,0)
left parenthesis Start Fraction 1 over 3 End Fraction comma 0 right parenthesis

(13,1)

Answer 3

Here are the responses needed to complete the proof that the medians of the triangle all meet at one point:

**Step 1:**
Put responses in the correct input:
- Draw each median so that it starts at a **vertex** and ends at the **midpoint** of the opposite side.

**Step 2:**
Find the coordinates of each midpoint by taking the sum of the x-coordinates and dividing by 2 and taking the sum of the y-coordinates and dividing by 2.

The midpoint between (0, 0) and (b, c) is **(b/2, c/2)**. The midpoint of (0, 0) and (a, 0) is **(a/2, 0)**. The midpoint of (a, 0) and (b, c) is **((a+b)/2, c/2)**.

**Step 3:**
Find the coordinates of P, the point where the medians appear to intersect, on each median. Use the formula **(1/3)(vertex) + (2/3)(midpoint)**.

**Step 4:**
Find the coordinates of P on the median that starts at vertex (0, 0) and ends at midpoint (a+b/2, c/2).

**(1/3)(0, 0) + (2/3)(a+b/2, c/2) = (0, 0) + (2/3)(a+b/2, c/2) = (2(a+b)/6, 2c/6) = (a+b)/3, c/3**

Find the coordinates of P on the median that starts at vertex (a, 0) and ends at midpoint (b/2, c/2).

**(1/3)(a, 0) + (2/3)(b/2, c/2) = (a/3, 0) + (b/3, c/3) = (a+b)/3, c/3**

Find the coordinates of P on the median that starts at vertex (b, c) and ends at midpoint (a/2, 0).

**(1/3)(b, c) + (2/3)(a/2, 0) = (b/3, c/3) + (a/3, 0) = (a+b)/3, c/3**

**Step 5:**
The coordinates of P on each median are **((a+b)/3, c/3)**, which proves that the three medians of this generic triangle all intersect at the same point.

Answer 4

To find the centroid of triangle ABC, we can use the formula for the centroid (G) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \):

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

Given the coordinates of the vertices:

- \( A(-2, -3) \)
- \( B(4, 0) \)
- \( C(-2, 2) \)

Now, we plug these coordinates into the formula:

**Calculate the x-coordinate of G:**
\[
x_G = \frac{-2 + 4 - 2}{3} = \frac{0}{3} = 0
\]

**Calculate the y-coordinate of G:**
\[
y_G = \frac{-3 + 0 + 2}{3} = \frac{-1}{3} = -\frac{1}{3}
\]

Thus, the coordinates of the centroid \( G \) are:
\[
G\left(0, -\frac{1}{3}\right)
\]

Checking the response options, the one that matches is:
- **(0, −1/3)**

Therefore, the centroid (which is also the point where the medians meet) is \( \left(0, -\frac{1}{3}\right) \), proving that the medians of triangle ABC meet at this single point.

Answer 5

To find the centroid of triangle ABC with vertices \( A(-3, -3) \), \( B(0, 6) \), and \( C(4, -3) \), we will use the centroid formula:

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

1. **Identify the coordinates of the vertices:**
- \( A(-3, -3) \)
- \( B(0, 6) \)
- \( C(4, -3) \)

2. **Calculate the x-coordinate of G:**

\[
x_G = \frac{-3 + 0 + 4}{3} = \frac{1}{3}
\]

3. **Calculate the y-coordinate of G:**

\[
y_G = \frac{-3 + 6 - 3}{3} = \frac{0}{3} = 0
\]

4. **Combine the coordinates of the centroid:**

Thus, the coordinates of the centroid \( G \) are:

\[
G\left(\frac{1}{3}, 0\right)
\]

From the provided responses, the matching answer is:

- **\(\left(\frac{1}{3}, 0\right)\)**

This proves that the medians of triangle ABC meet at the centroid at the point \(\left(\frac{1}{3}, 0\right)\).