To solve this problem, let's break it down step by step based on the given information in the image.

Problem:

• You have a line segment $PR$ with points $P(-6, 6)$ and $R(6, -6)$.
• Point $Q$ lies on segment $PR$.
• The ratio of $PQ$ to $QR$ is given as $\frac{PQ}{QR} = \frac{1}{2}$.

The goal is to find the coordinates of point $Q$.

Approach:

1. Using the section formula: The section formula allows us to find a point that divides a line segment in a given ratio. The formula for finding the coordinates of point $Q(x_1, y_1)$, which divides a line segment joining points $P(x_1, y_1)$ and $R(x_2, y_2)$ in the ratio $m:n$, is given by: $Q = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$ where $P(x_1, y_1) = (-6, 6)$, $R(x_2, y_2) = (6, -6)$, and the ratio $m:n = 1:2$.

2. Substitute the values into the formula: $x_Q = \frac{1 \times 6 + 2 \times (-6)}{1 + 2} = \frac{6 - 12}{3} = \frac{-6}{3} = -2$ $y_Q = \frac{1 \times (-6) + 2 \times 6}{1 + 2} = \frac{-6 + 12}{3} = \frac{6}{3} = 2$

Thus, the coordinates of $Q$ are $(-2, 2)$.

Final Answer:

The coordinates of $Q$ are $(-2, 2)$.

Would you like further details on how the section formula works, or any other explanations?

Relative Questions:

1. What is the section formula, and how is it derived?
2. How do you determine if a point divides a line externally or internally using ratios?
3. How can we apply the section formula to 3D coordinate geometry?
4. What is the difference between dividing a line segment in a ratio of $1:2$ versus $2:1$?
5. Can you find the midpoint of segment $PR$ using the midpoint formula?

Tip:

In coordinate geometry, always check whether the division of a segment is internal or external to ensure proper application of the section formula.