Let's analyze the math problem step by step based on the image you've uploaded.

1. Finding the slopes of $\overline{NP}$ and $\overline{TW}$

For $\overline{TW}$:

• The slope formula is given as: $m_{\overline{TW}} = -\frac{2}{1} = -2$

For $\overline{NP}$:

• The slope formula is given as: $m_{\overline{NP}} = \frac{5}{6}$

2. Are $\overline{NP}$ and $\overline{TW}$ parallel, perpendicular, or neither?

For two lines to be:

• Parallel: Their slopes must be equal.
• Perpendicular: The product of their slopes should equal $-1$.

Let's check:

• Slopes: $m_{\overline{NP}} = \frac{5}{6}$ and $m_{\overline{TW}} = -2$.
• The slopes are neither equal (so the lines are not parallel), nor is the product $-1$ (since $\frac{5}{6} \times (-2) = -\frac{10}{6}$, which is not $-1$).

So, $\overline{NP}$ and $\overline{TW}$ are neither parallel nor perpendicular.

3. Find the slope of the line perpendicular to $\overline{NP}$

• The slope of a line perpendicular to $\overline{NP}$ will be the negative reciprocal of $m_{\overline{NP}} = \frac{5}{6}$.
• The negative reciprocal of $\frac{5}{6}$ is $-\frac{6}{5}$.

So, the slope of the perpendicular line is $-\frac{6}{5}$.

4. Find the distance between points T and W

The coordinates of the points T and W from the graph are approximately:

• $T(3, 9)$
• $W(8, 0)$

To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the values: $d = \sqrt{(8 - 3)^2 + (0 - 9)^2} = \sqrt{(5)^2 + (-9)^2} = \sqrt{25 + 81} = \sqrt{106}$

Thus, the distance between points T and W is approximately: $d \approx 10.3 \text{ units}$

Would you like any further details on this, or have any other questions?

Here are five related questions to deepen your understanding:

1. What conditions must be met for two lines to be parallel?
2. How can you determine if two lines are perpendicular using their slopes?
3. Can you calculate the distance between two points if they lie on the same vertical line?
4. How does changing the slope of a line affect its steepness?
5. What is the geometric interpretation of the distance formula?

Tip: For lines, a reciprocal slope means flipping the fraction and changing the sign, which helps you easily find perpendicular slopes!