Question

Step 1: Draw the three medians of the triangle. Draw each median so that it starts at a and ends at the 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

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) + ((a + b)/3, c/3) = ((a + b)/3, c/3)

Find the coordinates of P on the median that starts at vertex

and ends at midpoint (b/2, c/2)

Answer 1

To find the coordinates of point \( P \) where the medians intersect, let's start by taking a systematic approach based on your steps.

### Step 1: Drawing the Medians of the Triangle
Assume we have a triangle defined by vertices:
- A at \( (0, 0) \)
- B at \( (a, 0) \)
- C at \( (b, c) \)

The midpoints of each side are:
- Midpoint \( D \) between B and C:
\[
D = \left(\frac{a + b}{2}, \frac{c}{2}\right)
\]
- Midpoint \( E \) between A and C:
\[
E = \left(\frac{b}{2}, \frac{c}{2}\right)
\]
- Midpoint \( F \) between A and B:
\[
F = \left(\frac{a}{2}, 0\right)
\]

The medians are drawn from each vertex to the midpoint of the opposite side:
1. From A (0,0) to D \(\left(\frac{a + b}{2}, \frac{c}{2}\right)\)
2. From B (a,0) to E \(\left(\frac{b}{2}, \frac{c}{2}\right)\)
3. From C (b,c) to F \(\left(\frac{a}{2}, 0\right)\)

### Step 2: Finding the Midpoints
You’ve already defined the midpoints correctly.

### Step 3: Find the Coordinates of P, the Centroid
The centroid \( P \) of a triangle is found by averaging the coordinates of the vertices:
\[
P = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
\]
Given \( A(0, 0) \), \( B(a, 0) \), \( C(b, c) \):
- \( P = \left(\frac{0 + a + b}{3}, \frac{0 + 0 + c}{3}\right) = \left(\frac{a + b}{3}, \frac{c}{3}\right) \)

### Step 4: Finding Coordinates of P on Each Median
Now let’s find point \( P \) on the median from A to D:
1. For the median from A to D:
\( P_A = \frac{1}{3} (0, 0) + \frac{2}{3} \left(\frac{a + b}{2}, \frac{c}{2}\right) \):
\[
P_A = \left(0 + \frac{2(a + b)}{6}, 0 + \frac{2c}{6}\right) = \left(\frac{a + b}{3}, \frac{c}{3}\right)
\]

2. For the median from B to E:
\( P_B = \frac{1}{3} (a, 0) + \frac{2}{3} \left(\frac{b}{2}, \frac{c}{2}\right) \):
\[
P_B = \left(\frac{1}{3}a + \frac{2}{3}\cdot\frac{b}{2}, \frac{1}{3} \cdot 0 + \frac{2}{3} \cdot \frac{c}{2}\right) = \left(\frac{1}{3}a + \frac{b}{3}, \frac{c}{3}\right) = \left(\frac{a + b}{3}, \frac{c}{3}\right)
\]

3. For the median from C to F:
\( P_C = \frac{1}{3} (b, c) + \frac{2}{3} \left(\frac{a}{2}, 0\right) \):
\[
P_C = \left(\frac{1}{3}b + \frac{2}{3}\cdot\frac{a}{2}, \frac{1}{3}c + \frac{2}{3}\cdot0\right) = \left(\frac{1}{3}b + \frac{a}{3}, \frac{c}{3}\right) = \left(\frac{a + b}{3}, \frac{c}{3}\right)
\]

### Conclusion
In all three cases, regardless of which median you analyze, you consistently arrive at the same centroid point:
\[
P = \left(\frac{a + b}{3}, \frac{c}{3}\right)
\]
Thus, the coordinates of \( P \) – the intersection point of the medians – indeed are \( P\left(\frac{a + b}{3}, \frac{c}{3}\right) \).