To solve this, let's break down each statement systematically and evaluate which ones are true.

Given Information:

• Coordinates for points:
  (-15.5, 0), (-5, -6.8), (4, -12), (0.4, -3), (-4, 8), (0, 6.2), (6.8, 13), (12, 0).

• Intersection Information:

  • Line q and r intersect at (6.8, 13).
  • Line p and r intersect at (0.4, -3).
  • Line m intersects the x-axis at (-15.5, 0) and the y-axis at (0, 6.2).
  • Line n intersects the x-axis at (12, 0).
  • The points (-4, 8) and (4, -12) lie on line p.
  • The point (-5, -6.8) lies on line n.

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Step 1: Determine the slope of each line.

For Line p:

• Points: (-4, 8) and (4, -12).
• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points (-4, 8) and (4, -12): $m_p = \frac{-12 - 8}{4 - (-4)} = \frac{-20}{8} = -2.5$

For Line q:

• Points: (-5, -6.8) and (6.8, 13).
• Using the same slope formula: $m_q = \frac{13 - (-6.8)}{6.8 - (-5)} = \frac{19.8}{11.8} \approx 1.68$

For Line r:

• Points: (0.4, -3) and (6.8, 13).
• Slope: $m_r = \frac{13 - (-3)}{6.8 - 0.4} = \frac{16}{6.4} = 2.5$

For Line m:

• Points: (-15.5, 0) and (0, 6.2).
• Slope: $m_m = \frac{6.2 - 0}{0 - (-15.5)} = \frac{6.2}{15.5} \approx 0.4$

For Line n:

• Points: (12, 0) and (0, 6.2).
• Slope: $m_n = \frac{6.2 - 0}{0 - 12} = \frac{6.2}{-12} \approx -0.52$

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Step 2: Evaluate the Statements.

A. p || q

• Lines p and q are not parallel because their slopes are different: $m_p = -2.5$ and $m_q \approx 1.68$.
  This statement is false.

B. q ⊥ n

• Two lines are perpendicular if the product of their slopes is -1.
• $m_q \times m_n \approx 1.68 \times (-0.52) = -0.8736$, which is not -1.
  This statement is false.

C. m || n

• Lines m and n are not parallel because their slopes are different: $m_m \approx 0.4$ and $m_n \approx -0.52$.
  This statement is false.

D. p ⊥ m

• To check if lines p and m are perpendicular, we need to check if the product of their slopes is -1.
• $m_p \times m_m = -2.5 \times 0.4 = -1$.
  This is exactly -1, so lines p and m are perpendicular.
  This statement is true.

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Final Answer:

The correct statement is:

• D. p ⊥ m

Would you like further clarification on any of these steps or calculations? Here are some additional questions related to the topic:

1. How do you determine if two lines are parallel or perpendicular using slopes?
2. What is the significance of the slope-intercept form in linear equations?
3. Can you find the equation of a line given two points?
4. How do you calculate the slope of a line from its graph?
5. What are some properties of parallel and perpendicular lines in geometry?

Tip: To quickly check if two lines are parallel, just compare their slopes. If the slopes are equal, the lines are parallel!