Sure! Let's find the algebraic and geometric mean coordinates of the points (3, 2), (1, 5), (-2, 3), and (0, 5).
### Step 1: Extract the Coordinates
We have the following coordinates:
- Point A: (3, 2)
- Point B: (1, 5)
- Point C: (-2, 3)
- Point D: (0, 5)
### Step 2: Separate X and Y Coordinates
For convenience, let's list the x-coordinates and y-coordinates separately:
- X-coordinates: 3, 1, -2, 0
- Y-coordinates: 2, 5, 3, 5
### Step 3: Calculate the Algebraic Mean Coordinates
#### Algebraic Mean of X Coordinates
To find the algebraic mean of the x-coordinates, sum them up and divide by the number of points:
[tex]\[
\text{Algebraic Mean of X} = \frac{3 + 1 - 2 + 0}{4} = \frac{2}{4} = 0.5
\][/tex]
#### Algebraic Mean of Y Coordinates
To find the algebraic mean of the y-coordinates, sum them up and divide by the number of points:
[tex]\[
\text{Algebraic Mean of Y} = \frac{2 + 5 + 3 + 5}{4} = \frac{15}{4} = 3.75
\][/tex]
Thus, the algebraic mean coordinates are [tex]\((0.5, 3.75)\)[/tex].
### Step 4: Calculate the Geometric Mean Coordinates
#### Geometric Mean of X Coordinates
To calculate the geometric mean of the x-coordinates, we use the formula for the geometric mean:
[tex]\[
\text{Geometric Mean of X} = (3 \times 1 \times -2 \times 0)^{\frac{1}{4}}
\][/tex]
Since one of the x-coordinates is 0, the product of the x-coordinates is 0. Therefore, the geometric mean of the x-coordinates is:
[tex]\[
\text{Geometric Mean of X} = 0
\][/tex]
#### Geometric Mean of Y Coordinates
To calculate the geometric mean of the y-coordinates, use the formula:
[tex]\[
\text{Geometric Mean of Y} = (2 \times 5 \times 3 \times 5)^{\frac{1}{4}}
\][/tex]
Calculating the product:
[tex]\[
2 \times 5 \times 3 \times 5 = 150
\][/tex]
Now calculate the fourth root of 150:
[tex]\[
(150)^{\frac{1}{4}} \approx 3.4996355115805833
\][/tex]
Thus, the geometric mean coordinates are [tex]\((0.0, 3.4996355115805833)\)[/tex].
### Summary
The algebraic mean coordinates are [tex]\((0.5, 3.75)\)[/tex] and the geometric mean coordinates are [tex]\((0.0, 3.4996355115805833)\)[/tex].
