Final answer:
Based on the given equation for line QR, it has a zero slope (horizontal line). To determine if QR is tangent to Circle O at R, we need to know the slope of line OR, which was not provided; hence, we cannot conclude the tangency relationship between QR and Circle O.
Step-by-step explanation:
The student's question pertains to whether the line with equation 5y = -40 + 41 is tangent to Circle O at point R(4,5). To answer this, we must first determine the slope of QR. Simplifying the equation of QR, we get y = -40/5 + 41/5, which simplifies to y = -8 + 8.2 or y = 0.2. This means QR has a slope of 0, since it is a horizontal line.
If QR is tangent to Circle O at point R, then the line passing through the center of the circle O and R (line OR) must be perpendicular to QR. Perpendicular lines have slopes that are negative reciprocals of each other. Since the slope of QR is 0, the slope of OR should be undefined (vertical line), meaning these two lines are perpendicular. However, we cannot confirm if OR is perpendicular to QR without additional information about the slope or equation of the line OR.
Therefore, based on the given information, we cannot definitively say whether QR is tangent to Circle O at point R without knowing the slope or orientation of OR relative to QR.