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.

M is located at (_,_)

Answer 1

To find point M on segment AB such that the ratio of AM to MB is 1:5, we can use the section formula.

Given:
- A = (1, 1)
- B = (7, -2)
- The ratio AM:MB = 1:5

Let's denote the coordinates of point M as (x, y). According to the section formula, the coordinates of point M can be calculated as follows:

\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m+n}
\]
\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m+n}
\]

where \( m \) and \( n \) are the parts of the ratio, \( (x_1, y_1) \) are the coordinates of point A, and \( (x_2, y_2) \) are the coordinates of point B.

Here, let \( m = 1 \) and \( n = 5 \):
- \( x_1 = 1 \), \( y_1 = 1 \)
- \( x_2 = 7 \), \( y_2 = -2 \)

Now, applying the section formula:

1. Calculate x-coordinate of M:
\[
x = \frac{1 \cdot 7 + 5 \cdot 1}{1 + 5} = \frac{7 + 5}{6} = \frac{12}{6} = 2
\]

2. Calculate y-coordinate of M:
\[
y = \frac{1 \cdot (-2) + 5 \cdot 1}{1 + 5} = \frac{-2 + 5}{6} = \frac{3}{6} = 0.5
\]

Thus, the coordinates of point M are \((2, 0.5)\).

Finally, rounding to the tenths place, \( M \) is located at:

**M is located at (2.0, 0.5)**.