In triangle PQR, with vertices P(5, 1), Q(0, 4), and R(2, 3), what is the equation of the altitude from P to the side QR?

Introduction / Context:
This question focuses on the equation of an altitude in a triangle in coordinate geometry. An altitude is a line from a vertex perpendicular to the opposite side. Finding its equation requires knowledge of slopes and perpendicularity.

Given Data / Assumptions:

- Triangle PQR with P(5, 1), Q(0, 4), and R(2, 3).
- PS is the altitude from P to side QR.
- We need the equation of line PS.

Concept / Approach:
To find the equation of an altitude from vertex P to side QR, we first find the slope of QR. Then we compute the negative reciprocal of that slope to get the slope of the altitude (because perpendicular lines have slopes whose product is −1). Finally, we use the point slope form with point P to write the equation of the altitude and simplify it.

Step-by-Step Solution:
Step 1: Compute the slope of QR. Q is (0, 4) and R is (2, 3), so the slope m_QR is (3 − 4) / (2 − 0) = −1 / 2.Step 2: The altitude from P to QR is perpendicular to QR, so its slope m_PS is the negative reciprocal of −1/2, which is 2.Step 3: Use the point slope form of a line with point P(5, 1) and slope 2: y − 1 = 2(x − 5).Step 4: Expand this equation: y − 1 = 2x − 10, so y = 2x − 9.Step 5: Rearrange into standard form: 2x − y − 9 = 0, or 2x − y = 9.

Verification / Alternative check:
Check that P(5, 1) lies on the line 2x − y = 9. Substituting x = 5 and y = 1 gives 2 * 5 − 1 = 10 − 1 = 9, which is true. Next, check that the slope of the altitude is indeed perpendicular to QR. QR has slope −1/2, while the altitude has slope 2, and their product is (−1/2) * 2 = −1, confirming perpendicularity.

Why Other Options Are Wrong:
- 2x + y = 9 has slope −2, which is not perpendicular to QR and also does not pass through P(5, 1) correctly.
- 7x + 2y = 33 and 2x − 7y = 9 have slopes −7/2 and 2/7 respectively, neither of which is the negative reciprocal of −1/2.
- x − 2y = −3 has slope 1/2, which is parallel to QR rather than perpendicular.

Common Pitfalls:
Students may confuse the concept of altitude with median or angle bisector, or forget that perpendicular lines have slopes that multiply to −1. Another common error is miscalculating slopes by swapping numerator and denominator or sign errors. Carefully computing slopes and using the point slope form avoids these issues.

Final Answer:
The equation of altitude PS is 2x − y = 9.