Question

the coordinates of the endpoints of NO are N(-9,1) and O(7,9) point P is on NO and divides it such that NP:OP is 1:3 \What are the coordinates of P
Write your answer as integers or Decimal

Answer 1

To find the coordinates of point P that divides the line segment NO in the ratio \( NP:OP = 1:3 \), we can use the section formula.

The coordinates of points N and O are:
- \( N(-9, 1) \)
- \( O(7, 9) \)

Let \( P(x, y) \) be the point dividing the segment in the ratio \( m:n = 1:3 \). According to the section formula, the coordinates of point P can be calculated as follows:

\[
x = \frac{mx_2 + nx_1}{m+n}
\]
\[
y = \frac{my_2 + ny_1}{m+n}
\]

where \( (x_1, y_1) \) are the coordinates of point N, and \( (x_2, y_2) \) are the coordinates of point O.

Substituting the values:
- \( m = 1 \)
- \( n = 3 \)
- \( (x_1, y_1) = (-9, 1) \)
- \( (x_2, y_2) = (7, 9) \)

### Calculate the coordinates of P:
**For x-coordinate:**
\[
x = \frac{1 \cdot 7 + 3 \cdot (-9)}{1 + 3} = \frac{7 - 27}{4} = \frac{-20}{4} = -5
\]

**For y-coordinate:**
\[
y = \frac{1 \cdot 9 + 3 \cdot 1}{1 + 3} = \frac{9 + 3}{4} = \frac{12}{4} = 3
\]

### Final coordinates of point P:
Thus, the coordinates of point P are:
\[
P(-5, 3)
\]