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.(1 point) M is located at (,).

Answer 1

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

The coordinates of M can be found using the formula:

\[ M = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \]

where \(A(x_1, y_1) = (1, 1)\), \(B(x_2, y_2) = (7, -2)\), and \(m:n = 1:5\) means \(m = 1\) and \(n = 5\).

Substituting in the values:

\[ M_x = \frac{1 \cdot 7 + 5 \cdot 1}{1+5} = \frac{7 + 5}{6} = \frac{12}{6} = 2 \]

\[ M_y = \frac{1 \cdot (-2) + 5 \cdot 1}{1+5} = \frac{-2 + 5}{6} = \frac{3}{6} = 0.5 \]

Thus, the coordinates of point M are:

\[ M(2, 0.5) \]

To the tenths place, M is located at \((2.0, 0.5)\).