Given a right triangle with vertices at $(0,0)$, $(4,0)$, and $(4,3)$, let point $z$ be $(4,0)$ and line $ab$ be the segment from $(0,0)$ to $(4,0)$. Which of the following points lies on the line that passes through point $z$ and is perpendicular to line $ab$?

• A. $(0,3)$
• B. $(4,2)$
• C. $(3,0)$
• D. $(2,0)$

Answer: (B)

Verified step by step guidance

1. Identify the coordinates of points \( a \) and \( b \). Here, \( a = (0,0) \) and \( b = (4,0) \).
2. Determine the slope of line segment \( ab \). Since both points have the same \( y \)-coordinate, the slope \( m_{ab} = \frac{0 - 0}{4 - 0} = 0 \).
3. Find the slope of the line perpendicular to \( ab \). The perpendicular slope \( m_{\perp} \) is the negative reciprocal of \( m_{ab} \). Since \( m_{ab} = 0 \), the perpendicular slope is undefined, indicating a vertical line.
4. Write the equation of the line passing through point \( z = (4,0) \) with the perpendicular slope. Because the slope is undefined, the line is vertical and has the equation \( x = 4 \).
5. Check which of the given points satisfy the equation \( x = 4 \). The point(s) with \( x = 4 \) lie on the perpendicular line through \( z \).