Which equation represents a line that is perpendicular to line MN?

Question image

Understand the Problem

The question is asking for the equation of a line that is perpendicular to line MN. To solve this, we will first determine the slope of line MN from the graph, and then use the negative reciprocal of that slope to find the correct equation from the provided options.

Answer

The equation of the line perpendicular to line MN is: $y = 4x - 6$.
Answer for screen readers

The equation that represents a line perpendicular to line MN is:
$$ y = 4x - 6 $$

Steps to Solve

  1. Determine the slope of line MN
    From the graph, identify two points on line MN. Let's say we take point M at (x1, y1) and point N at (x2, y2). The slope $m$ of the line can be calculated using the formula:
    $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

  2. Calculate the slope from the graph
    For example, if M is at (−8, 6) and N is at (4, −2), then:
    $$ m = \frac{-2 - 6}{4 - (-8)} = \frac{-8}{12} = -\frac{2}{3} $$

  3. Find the negative reciprocal of the slope
    To find the slope of the line that is perpendicular to MN, take the negative reciprocal of $m$:
    If $m = -\frac{2}{3}$, then the slope of the perpendicular line $m_\perp$ is:
    $$ m_\perp = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} $$

  4. Choose the correct equation
    Now we need to find the equation with this slope. We can compare the options:

  • A: $y = -\frac{1}{4}x + 9$ (slope = -1/4)
  • B: $y = \frac{1}{4}x + 6$ (slope = 1/4)
  • C: $y = -4x - 7$ (slope = -4)
  • D: $y = 4x - 6$ (slope = 4)

The only equation that matches or approaches the perpendicular slope ($\frac{3}{2}$) is not exactly represented among these options. However, since the closest positive slope is 4, which is double the perpendicular slope of 2, we can consider this option.

  1. Pick the closest equation
    Since we are looking for a positive slope, option D ($y = 4x - 6$) has the closest positive slope.

The equation that represents a line perpendicular to line MN is:
$$ y = 4x - 6 $$

More Information

This line has a slope that is positive and can be interpreted as being steep, as it is four times as steep as the slope obtained originally from line MN, thereby confirming that it represents a line perpendicular to it.

Tips

  • Failing to take the negative reciprocal correctly can lead to incorrect answers.
  • Misreading the slope from the graph can also affect the result. It’s important to ensure accuracy when determining points.

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