Question

the coordinates of the endpoints of MN are M(-7,-6) and N(1,2) point O is on MN and divides it such that MO:NO is 3:1
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 \( MO:NO = 3:1 \), we can use the section formula. Given the coordinates \( M(-7, -6) \) and \( N(1, 2) \), we can use the formula for the coordinates of a point dividing a segment in the ratio \( m:n \):

\[
O_x = \frac{n \cdot M_x + m \cdot N_x}{m+n}
\]
\[
O_y = \frac{n \cdot M_y + m \cdot N_y}{m+n}
\]

In this case, \( m = 3 \) and \( n = 1 \).

### Step 1: Calculate \( O_x \)
Using the x-coordinates:
\[
O_x = \frac{1 \cdot (-7) + 3 \cdot 1}{3 + 1} = \frac{-7 + 3}{4} = \frac{-4}{4} = -1
\]

### Step 2: Calculate \( O_y \)
Using the y-coordinates:
\[
O_y = \frac{1 \cdot (-6) + 3 \cdot 2}{3 + 1} = \frac{-6 + 6}{4} = \frac{0}{4} = 0
\]

### Step 3: Combine results
Thus, the coordinates of \( O \) are:
\[
O(-1, 0)
\]

Therefore, the coordinates of point \( O \) are \(\boxed{(-1, 0)}\).