Question

The coordinates of the endpoints of OP are O(9,1) and P(18,19) points Q is on OP and divides it such that OQ:PQ is 5:4
what are th coordinates of Q
Write your answers as integers or decimals

Answer 1

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:

1. Calculate the x-coordinate of Q:
\[
x_Q = \frac{5 \cdot 18 + 4 \cdot 9}{5 + 4} = \frac{90 + 36}{9} = \frac{126}{9} = 14
\]

2. 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 \((14, 11)\).