Question

find the centroid in this formula: x1+x2+x3/3, y1+y2+y3/3)

(3, 14/3)
(3, 5)
(3, 6)
(10/3, 14/3)

Answer 1

To find the centroid (geometric center) of the given points using the formula you provided, we need to average the x-coordinates and the y-coordinates of all the points.

The formula for finding the centroid \((C_x, C_y)\) of three points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is:

\[
C_x = \frac{x_1 + x_2 + x_3}{3}, \quad C_y = \frac{y_1 + y_2 + y_3}{3}
\]

Let's calculate the centroid using the given points:

1. **First set of points**: (3, \(\frac{14}{3}\)), (3, 5), (3, 6)

**Step 1: Calculate the x-coordinate of the centroid**:
\[
C_x = \frac{3 + 3 + 3}{3} = \frac{9}{3} = 3
\]

**Step 2: Calculate the y-coordinate of the centroid**:
\[
C_y = \frac{\frac{14}{3} + 5 + 6}{3}
\]

Convert 5 and 6 to fractions with a denominator of 3:
\[
5 = \frac{15}{3}, \quad 6 = \frac{18}{3}
\]
Now plug these into \(C_y\):
\[
C_y = \frac{\frac{14}{3} + \frac{15}{3} + \frac{18}{3}}{3} = \frac{\frac{14 + 15 + 18}{3}}{3} = \frac{\frac{47}{3}}{3} = \frac{47}{9}
\]

Hence, the centroid of the first three points is \((3, \frac{47}{9})\).

2. **Second set of points**: (10/3, \(\frac{14}{3}\))

Now, we can find the centroid of all four points by using the same formula for the four points.

**Step 3: Calculate the x-coordinate of the combined centroid**:
\[
C_x = \frac{3 + 3 + 3 + \frac{10}{3}}{4} = \frac{9 + \frac{10}{3}}{4} = \frac{\frac{27}{3} + \frac{10}{3}}{4} = \frac{\frac{37}{3}}{4} = \frac{37}{12}
\]

**Step 4: Calculate the y-coordinate of the combined centroid**:
\[
C_y = \frac{\frac{14}{3} + 5 + 6 + \frac{14}{3}}{4}
\]
Combine the y-coordinates as before:
\[
C_y = \frac{\frac{14}{3} + \frac{15}{3} + \frac{18}{3} + \frac{14}{3}}{4} = \frac{\frac{14 + 15 + 18 + 14}{3}}{4} = \frac{\frac{61}{3}}{4} = \frac{61}{12}
\]

Thus, the overall centroid of the given four points is:

\[
(C_x, C_y) = \left(\frac{37}{12}, \frac{61}{12}\right)
\]