Final answer:
The equations of the two sides of square PORS passing through vertex P are y = -3x + 9 (side OP) and x = 2 (side PS).
Step-by-step explanation:
The student asked to find the equation of two sides of square PORS through vertex P(2, 3) given that the equation of diagonal PR is x - 3y = 2. To find the equations of the sides OP and PS, we must first understand that these sides are perpendicular to the diagonal PR and thus will have slopes that are negative reciprocals of the slope of PR. Since the diagonal PR has a slope of 1/3 (from the equation of the line x - 3y = 2), the slopes of sides OP and PS must be -3.
We use point-slope form, y - y1 = m(x - x1), to find the equation of side OP which passes through P(2, 3). Substituting in P's coordinates (x1 = 2, y1 = 3) and the slope m = -3, we get the equation y - 3 = -3(x - 2). Simplifying this, we obtain y = -3x + 9 as the equation for side OP.
Since square PORS is symmetric, side PS will be parallel to side OR and perpendicular to side OP. Hence, it will also have a slope of -3. To find the equation, we again use P's coordinates and the slope of -3 to write the equation for side PS: y - 3 = -3(x - 2), resulting again in y = -3x + 9. However, side PS is a vertical line because it must be perpendicular to the horizontal side OP, hence its equation is of the form x = a. Since it passes through P(2, 3), its equation must be x = 2.
Therefore, the equations of sides OP and PS are y = -3x + 9 and x = 2, respectively.