Question

Find a point, P, that is 1/6 of the way from point A to point B.

A (-2,4)
B (4,2)

a. (0,3)
b. (0,4)
c. (-1.5,3)
d. (-1,3.67)

Answer 1

To find a point \( P \) that is \( \frac{1}{6} \) of the way from point \( A \) to point \( B \), we can use the following formula for finding a point \( P \) that divides the line segment between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \):

\[
P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

In this case, the ratio \( m:n \) is \( 1:5 \) because \( \frac{1}{6} \) of the way from \( A \) to \( B \) means \( 1 \) part toward \( B \) and the remaining \( 5 \) parts toward \( A \).

Given:
- \( A(-2, 4) \) (where \( x_1 = -2 \) and \( y_1 = 4 \))
- \( B(4, 2) \) (where \( x_2 = 4 \) and \( y_2 = 2 \))

Now, we can substitute \( m = 1 \) and \( n = 5 \) into the formula:

1. Calculate \( P_x \):

\[
P_x = \frac{1 \cdot 4 + 5 \cdot (-2)}{1 + 5} = \frac{4 - 10}{6} = \frac{-6}{6} = -1
\]

2. Calculate \( P_y \):

\[
P_y = \frac{1 \cdot 2 + 5 \cdot 4}{1 + 5} = \frac{2 + 20}{6} = \frac{22}{6} = \frac{11}{3} \approx 3.67
\]

So, the coordinates of point \( P \) are:

\[
P \left(-1, \frac{11}{3}\right) \approx (-1, 3.67)
\]

Thus, the correct answer is:

**d. (-1, 3.67)**.