Question

1. are lines n and k perpendicular?

Explanation:

Step1: Find slope of line n

First, identify two points on line n. Let's take points from the graph. For line n, let's say it passes through (0, 4) and (4, -4) (from the graph). The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. So, $m_n=\frac{-4 - 4}{4 - 0}=\frac{-8}{4}=-2$.

Step2: Find slope of line k

For line k, let's take points (4, 0) and (2, -2) (from the graph). Using the slope formula, $m_k=\frac{-2 - 0}{2 - 4}=\frac{-2}{-2}=1$? Wait, no, maybe better points. Wait, line k: let's see, another pair. Let's take (4,0) and (3, -1)? Wait, maybe I made a mistake. Wait, line k: from the graph, let's check the coordinates. Wait, line k goes through (4,0) and (2, -2)? Wait, no, (4,0) to (3, -1) is not right. Wait, maybe line k passes through (4,0) and (2, -2)? Wait, no, let's re - examine. Wait, line k: let's take two clear points. Let's say line k passes through (4, 0) and (3, -1)? No, maybe (4,0) and (2, -2): the slope would be $\frac{-2 - 0}{2 - 4}=\frac{-2}{-2}=1$. Wait, but line n: let's take (0,4) and (4, -4), slope is -2. Wait, no, maybe I got the lines wrong. Wait, line n: let's see, the red line n. Let's take two points: (0,4) and (4, -4)? Wait, no, maybe ( - 2,0) and (2, - 4)? Wait, slope would be $\frac{-4 - 0}{2 - (-2)}=\frac{-4}{4}=-1$. Wait, maybe I misidentified the lines. Wait, let's start over.

Wait, line n: let's find two points. Let's say line n passes through (0, 4) and (4, 0)? No, that's line g. Wait, line n: the red line that goes down from left to right. Let's take ( - 2, 0) and (2, - 4). So $x_1=-2,y_1 = 0,x_2 = 2,y_2=-4$. Then slope $m_n=\frac{-4 - 0}{2-(-2)}=\frac{-4}{4}=-1$.

Line k: the red line going up from left to right. Let's take (2, - 2) and (4, 0). So $x_1 = 2,y_1=-2,x_2 = 4,y_2 = 0$. Then slope $m_k=\frac{0 - (-2)}{4 - 2}=\frac{2}{2}=1$.

Step3: Check if slopes are negative reciprocals

Two lines are perpendicular if the product of their slopes is - 1. So $m_n\times m_k=(-1)\times(1)=-1$.

Answer:

Yes, lines n and k are perpendicular because the product of their slopes is - 1.