Question

Please help
Using the image,find a point, P, that is 1/6 of the way from point A to point B.
Point A is at -2,4 point B is at 4,2

Answer 1

Can you please explain further the x coordinate of p and the y coordinate of p

Answer 2

To find a point \( P \) that is \( \frac{1}{6} \) of the way from point \( A \) to point \( B \), we can use the formula for finding a point along a line segment between two points.

1. **Identify the coordinates of points A and B:**
- Point \( A \): \( (-2, 4) \)
- Point \( B \): \( (4, 2) \)

2. **Calculate the difference in coordinates:**
- The change in the x-coordinate:
\[
\Delta x = x_B - x_A = 4 - (-2) = 4 + 2 = 6
\]
- The change in the y-coordinate:
\[
\Delta y = y_B - y_A = 2 - 4 = -2
\]

3. **Calculate \( \frac{1}{6} \) of the change in coordinates:**
- Change in x to point P:
\[
\frac{1}{6} \Delta x = \frac{1}{6} \times 6 = 1
\]
- Change in y to point P:
\[
\frac{1}{6} \Delta y = \frac{1}{6} \times (-2) = -\frac{1}{3}
\]

4. **Calculate the coordinates of point P:**
- The x-coordinate of \( P \):
\[
x_P = x_A + \frac{1}{6} \Delta x = -2 + 1 = -1
\]
- The y-coordinate of \( P \):
\[
y_P = y_A + \frac{1}{6} \Delta y = 4 - \frac{1}{3} = 4 - 0.333... \approx 3.67
\]

Thus, the coordinates of point \( P \) are approximately:
\[
P = (-1, \frac{11}{3}) \quad \text{or} \quad P \approx (-1, 3.67)
\]

So, the point \( P \) that is \( \frac{1}{6} \) of the way from \( A \) to \( B \) is:
\[
P \approx (-1, 3.67)
\]

Answer 3

Certainly! Let's break down the calculation of the x-coordinate and y-coordinate of point \( P \) in more detail.

### Given Points
- Point \( A = (-2, 4) \)
- Point \( B = (4, 2) \)

### Step 1: Calculate the Differences in Coordinates
We need to find how far apart points \( A \) and \( B \) are in both the x and y directions.

1. **Change in X coordinates**
\[
\Delta x = x_B - x_A = 4 - (-2) = 4 + 2 = 6
\]

2. **Change in Y coordinates**
\[
\Delta y = y_B - y_A = 2 - 4 = -2
\]

### Step 2: Find \( \frac{1}{6} \) of the Way Between A and B
Since we need to find point \( P \) that is \( \frac{1}{6} \) of the way from \( A \) to \( B \), we calculate:

1. **For the x-coordinate:**
- Start by calculating \( \frac{1}{6} \) of the \( \Delta x \):
\[
\frac{1}{6} \Delta x = \frac{1}{6} \times 6 = 1
\]

- Now, add this change to the x-coordinate of point \( A \):
\[
x_P = x_A + \frac{1}{6} \Delta x = -2 + 1 = -1
\]

2. **For the y-coordinate:**
- Start by calculating \( \frac{1}{6} \) of the \( \Delta y \):
\[
\frac{1}{6} \Delta y = \frac{1}{6} \times (-2) = -\frac{1}{3}
\]

- Now, add this change to the y-coordinate of point \( A \):
\[
y_P = y_A + \frac{1}{6} \Delta y = 4 - \frac{1}{3}
\]
To perform the subtraction, convert \( 4 \) to a fraction:
\[
4 = \frac{12}{3}
\]
Now subtract:
\[
y_P = \frac{12}{3} - \frac{1}{3} = \frac{11}{3}
\]

### Final Coordinates of Point \( P \)
Putting it all together, point \( P \) is at:
\[
P = (-1, \frac{11}{3}) \quad \text{or} \quad P \approx (-1, 3.67)
\]

Thus, the coordinates of point \( P \) are:
- **x-coordinate:** \( -1 \)
- **y-coordinate:** \( \frac{11}{3} \), which is approximately \( 3.67 \)

This explanation clarifies how we found each coordinate of point \( P \) step by step. If you have any more questions or need further details, feel free to ask!