The graph in the image shows two lines, one red and one blue, and asks whether the lines are parallel, perpendicular, or neither. Let's analyze the slopes of the lines to answer the question.

Step 1: Identify the Slopes

We can calculate the slope ($m$) of a line using two points on the line with the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• For the red line, it passes through points $(-5, -5)$ and $(0, 0)$, so: $m_{\text{red}} = \frac{0 - (-5)}{0 - (-5)} = \frac{5}{5} = 1$
• For the blue line, it passes through points $(-5, -2)$ and $(0, 0)$, so: $m_{\text{blue}} = \frac{0 - (-2)}{0 - (-5)} = \frac{2}{5}$

Step 2: Analyze the Relationship Between the Slopes

• If the slopes are equal ($m_1 = m_2$), the lines are parallel.
• If the product of the slopes is $-1$ ($m_1 \cdot m_2 = -1$), the lines are perpendicular.
• If neither condition is met, the lines are neither parallel nor perpendicular.

Now, let's check: $m_{\text{red}} \cdot m_{\text{blue}} = 1 \cdot \frac{2}{5} = \frac{2}{5}$ Since $\frac{2}{5} \neq -1$ and the slopes are not equal, the lines are neither parallel nor perpendicular.

Would you like further details or have any questions about this?

Related Questions:

1. How are slopes used to determine if two lines are parallel?
2. What condition must be met for two lines to be perpendicular?
3. Can lines with slopes that aren't equal or negative reciprocals ever be parallel or perpendicular?
4. What is the significance of slope in graphing linear equations?
5. How can you quickly estimate the slope of a line just by looking at a graph?

Tip: Remember, perpendicular lines have slopes that are negative reciprocals, which means their product is always $-1$.