To find the coordinates of point Q that divides the line segment OP in the ratio of 5:4, we can use the section formula. The section formula states that if a point divides the line segment joining two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of the point dividing the segment can be calculated as follows:
\[ Q\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In this case:
- Let \(O\) be \(A(9, 1)\) and \(P\) be \(B(18, 19)\).
- The ratio \(OQ:PQ\) is \(5:4\), meaning \(m = 5\) and \(n = 4\).
Now, we will apply the section formula:
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Calculate the x-coordinate of Q: \[ x_Q = \frac{5 \cdot 18 + 4 \cdot 9}{5 + 4} = \frac{90 + 36}{9} = \frac{126}{9} = 14 \]
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Calculate the y-coordinate of Q: \[ y_Q = \frac{5 \cdot 19 + 4 \cdot 1}{5 + 4} = \frac{95 + 4}{9} = \frac{99}{9} = 11 \]
Therefore, the coordinates of point Q are \((14, 11)\).