Straight Lines
Foot of Perpendicular
Premium Question
Grade 11
Question:
The foot of the perpendicular from the point $(\alpha, \beta)$ to the line $y = x$ is:
(1) $(\alpha, \alpha)$
(2) $\left(\frac{\alpha+\beta}{2}, \frac{\alpha+\beta}{2}\right)$
(3) $(\beta, \beta)$
(4) $\left(\frac{\alpha-\beta}{2}, \frac{\alpha-\beta}{2}\right)$
Step-by-Step Solution
Key Concept: The foot $(h, k)$ lies on $y = x$ (so $h = k$) and $(\alpha, \beta) \to (h, k)$ is perpendicular to $y = x$ (slope of the segment is $-1$). Set up two equations and solve.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)