Question

if line sm passes through points s(0,0) and m(2,4), and line sd passes through points s(0,0) and d(4, - 2), are lines sm and sd perpendicular?
no, because their slopes are neither opposite reciprocals nor the same.
yes, because their slopes are opposite reciprocals.
no, there is not enough information to determine if the lines are perpendicular.
yes, because their slopes are the same.

Explanation:

Step1: Calculate slope of line SM

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $S(0,0)$ and $M(2,4)$, $m_{SM}=\frac{4 - 0}{2 - 0}=\frac{4}{2}=2$.

Step2: Calculate slope of line SD

For points $S(0,0)$ and $D(4,-2)$, $m_{SD}=\frac{-2 - 0}{4 - 0}=\frac{-2}{4}=-\frac{1}{2}$.

Step3: Check perpendicular - slope relationship

Two lines are perpendicular if the product of their slopes is - 1, i.e., the slopes are opposite reciprocals. Here, $m_{SM}\times m_{SD}=2\times(-\frac{1}{2})=-1$.

Answer:

Yes, because their slopes are opposite reciprocals.