The problem in the image translates to:

"If the points $P(7, 3)$, $Q(-3, 1)$, and $R(2, -3)$ lie in a Cartesian plane, what is the position of point $Q$ relative to the line PR?"

Solution:

We can determine the position of point $Q$ relative to line $PR$ by checking if it lies on the line, is above the line, or is below the line.

1. Equation of Line PR: To find the equation of line PR, we first calculate the slope $m$ using points $P(7, 3)$ and $R(2, -3)$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 3}{2 - 7} = \frac{-6}{-5} = \frac{6}{5}$

   Using the point-slope form of the equation of a line, $y - y_1 = m(x - x_1)$, and substituting $P(7, 3)$ and $m = \frac{6}{5}$: $y - 3 = \frac{6}{5}(x - 7)$ Expanding: $y - 3 = \frac{6}{5}x - \frac{42}{5}$ $y = \frac{6}{5}x - \frac{42}{5} + 3 = \frac{6}{5}x - \frac{42}{5} + \frac{15}{5} = \frac{6}{5}x - \frac{27}{5}$ So, the equation of line PR is: $y = \frac{6}{5}x - \frac{27}{5}$

2. Position of Point Q: Now, substitute the coordinates of point $Q(-3, 1)$ into the equation of line PR: $y = \frac{6}{5}(-3) - \frac{27}{5} = \frac{-18}{5} - \frac{27}{5} = \frac{-45}{5} = -9$

   Since the y-coordinate of point $Q$ is 1, and the value of $y$ on the line PR when $x = -3$ is $-9$, point $Q$ does not lie on the line and is above the line because $1 > -9$.

Answer:

Point $Q$ is above the line PR.

Would you like further details or have any questions? Here are five related questions to deepen your understanding:

1. How would you determine the equation of a line passing through two different points?
2. What does the slope of a line represent geometrically?
3. How can we verify whether a point lies on a specific line?
4. How do you calculate the perpendicular distance from a point to a line?
5. What other methods can you use to find the relative position of a point to a line?

Tip:

For any line in the Cartesian plane, the relative position of a point can be determined by comparing the y-coordinate of the point with the value given by the line equation at the same x-coordinate.