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 \( (13, 12) \).
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