Question

The coordinates of the endpoints of NO are N (5,16) and O(19,9) point is on NO and divides it such that NP:OP is 4:3
what are the coordinates of P
Write your answer as integers or decimals

Answer 1

To find the coordinates of point \( P \) that divides the line segment \( NO \) in the ratio \( 4:3 \), we can use the section formula.

Given the coordinates of points \( N(5, 16) \) and \( O(19, 9) \), we can denote \( P \) as dividing \( NO \) so that \( NP:OP = 4:3 \). This means that \( P \) divides the segment into parts that form the ratio \( m:n = 4:3 \).

The section formula for a point dividing a segment in the ratio \( m:n \) is given by:

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

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

Plugging in the values:

- \( m = 4 \)
- \( n = 3 \)
- \( x_1 = 5 \)
- \( y_1 = 16 \)
- \( x_2 = 19 \)
- \( y_2 = 9 \)

Now, substituting these values into the formula gives:

\[
P_x = \frac{4 \cdot 19 + 3 \cdot 5}{4 + 3} = \frac{76 + 15}{7} = \frac{91}{7} = 13
\]

\[
P_y = \frac{4 \cdot 9 + 3 \cdot 16}{4 + 3} = \frac{36 + 48}{7} = \frac{84}{7} = 12
\]

Therefore, the coordinates of point \( P \) are \( (13, 12) \).