Question

the coordinates of the endpoints of MN are M(-11,8) and N(73,-6) point O is on MN and divides it such that MO:NO is 1:6 What are the coordinates of O
Write your answer as integers or Decimal

Answer 1

To find the coordinates of point \( O \) that divides the line segment \( MN \) in the ratio \( 1:6 \), we can use the section formula.

Given the coordinates of points \( M \) and \( N \):

- \( M(-11, 8) \)
- \( N(73, -6) \)

Let \( O \) divide \( MN \) in the ratio \( 1:6 \). Thus, \( MO:ON = 1:6 \).

Using the section formula:

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

Where:
- \( m = 1 \) (the part of the ratio we are dividing),
- \( n = 6 \),
- \( (x_1, y_1) = (-11, 8) \) (coordinates of \( M \)),
- \( (x_2, y_2) = (73, -6) \) (coordinates of \( N \)).

Now substituting the values into the formula:

For the x-coordinate:

\[
x_O = \frac{1 \cdot 73 + 6 \cdot (-11)}{1 + 6} = \frac{73 - 66}{7} = \frac{7}{7} = 1
\]

For the y-coordinate:

\[
y_O = \frac{1 \cdot (-6) + 6 \cdot 8}{1 + 6} = \frac{-6 + 48}{7} = \frac{42}{7} = 6
\]

Thus, the coordinates of point \( O \) are \( (1, 6) \).

Therefore, the coordinates of point \( O \) are:

\[
\boxed{(1, 6)}
\]