Question

Given points A(1,1) and B(7,-2) determine the coordinates of point M on AB such that the ratio of AM to MB is 1;5. write your responses as decimal values if necessary to the tenths place.

Answer 1

To find the coordinates of point M that divides the line segment AB in the ratio of 1:5, we can use the section formula.

Given:
- A(1, 1)
- B(7, -2)
- The ratio \( k_1:k_2 = 1:5 \)

The section formula for dividing a line segment in the ratio \( m:n \) is given by:

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

In our case, \( m = 1 \) and \( n = 5 \):

\[
x_1 = 1, \quad y_1 = 1 \quad (coordinates \ of \ A)
\]
\[
x_2 = 7, \quad y_2 = -2 \quad (coordinates \ of \ B)
\]

Now substituting these values into the section formula:

\[
M\left(\frac{1 \cdot 7 + 5 \cdot 1}{1+5}, \frac{1 \cdot (-2) + 5 \cdot 1}{1+5}\right)
\]

Calculating the x-coordinate of M:

\[
M_x = \frac{7 + 5}{6} = \frac{12}{6} = 2
\]

Calculating the y-coordinate of M:

\[
M_y = \frac{-2 + 5}{6} = \frac{3}{6} = 0.5
\]

Therefore, the coordinates of point M are:

\[
M(2, 0.5)
\]

So, the final answer is:

\[
M(2.0, 0.5)
\]