Question

The coordinates of the endpoints of OP are O(-12,8) and P(0,-4) Point Q is on OP and divides it such the OQ:PQ is 1:3
what are the coordinates of Q
write your answer as integers or decimals

Answer 1

To find the coordinates of point Q that divides the line segment OP in the ratio 1:3, we can use the section formula. The section formula states that if a point divides a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), the coordinates of the point \( P(x, y) \) can be calculated as:

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

Here, point O has coordinates \( O(-12, 8) \) and point P has coordinates \( P(0, -4) \), and the ratio \( OQ:PQ = 1:3 \). Thus, we have \( m = 1 \) and \( n = 3 \).

Now we can substitute into the formulas:

1. **Calculating the x-coordinate of Q:**

\[
x_Q = \frac{1 \cdot 0 + 3 \cdot (-12)}{1 + 3} = \frac{0 - 36}{4} = \frac{-36}{4} = -9
\]

2. **Calculating the y-coordinate of Q:**

\[
y_Q = \frac{1 \cdot (-4) + 3 \cdot 8}{1 + 3} = \frac{-4 + 24}{4} = \frac{20}{4} = 5
\]

Therefore, the coordinates of point Q are:

\[
Q(-9, 5)
\]

Thus, the final answer is:

\[
\text{Coordinates of Q: } (-9, 5)
\]