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The location of runway A for a new airport is shown on the coordinate plane.

Two additional runways are planned. Each will pass through the point (0, -4) on the coordinate plane.

Runway B is to be parallel to runway A.

Runway C is to be perpendicular to runway A.

Which equations could describe either of those runways? Choose all that are correct.

(2 points)
Responses

y=−43x−4
y is equal to negative 4 thirds x minus 4

y=−34x−4
y is equal to negative 3 fourths x minus 4

y=34x−4
y is equal to 3 fourths x minus 4

y=43x−4
y is equal to 4 thirds x minus 4
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1 answer

To determine the equations that could describe runways B and C, we need to know the slope of runway A. However, since the slope of runway A is not provided in the question, I'll demonstrate how to find parallel and perpendicular lines based on the general concept.

  1. Parallel Line (Runway B): A line that is parallel to another has the same slope. If the slope of runway A is \(m\), then the slope of runway B will also be \(m\). The equation of Runway B can be expressed in point-slope form as: \[ y + 4 = m(x - 0) \implies y = mx - 4 \]

  2. Perpendicular Line (Runway C): A line that is perpendicular to another line has a slope that is the negative reciprocal of the original slope. If the slope of runway A is \(m\), then the slope of runway C will be \(-\frac{1}{m}\). The equation of Runway C can be expressed similarly: \[ y + 4 = -\frac{1}{m}(x - 0) \implies y = -\frac{1}{m} x - 4 \]

Analysis of Given Equations

Now let's analyze the given response options in terms of potential slopes that fit the description of parallel or perpendicular lines based on a hypothetical slope \(m\) for runway A.

  • \(y = -\frac{4}{3}x - 4\): This line has a slope of \(-\frac{4}{3}\).
  • \(y = -\frac{3}{4}x - 4\): This line has a slope of \(-\frac{3}{4}\).
  • \(y = \frac{3}{4}x - 4\): This line has a slope of \(\frac{3}{4}\).
  • \(y = \frac{4}{3}x - 4\): This line has a slope of \(\frac{4}{3}\).

Conclusion

Without knowing the slope of runway A, we theoretically cannot assert which of the equations represent the runways. However, if we assume a slope for runway A:

  • If runway A has a slope of 1 (for example), runways B and C could logically have slopes (1) and (-1) respectively.
  • If the slope is \(-\frac{4}{3}\), then B would be parallel \(-\frac{4}{3}\) and C would be \(\frac{3}{4}\).

In the latter example, this means among the given equations, only the options with \(-\frac{4}{3}\) and \(\frac{3}{4}\) could justify.

Answer

  • \(y = -\frac{4}{3}x - 4\) (could be Runway B if A's slope is \(-\frac{4}{3}\))
  • \(y = \frac{3}{4}x - 4\) (could be Runway C if A's slope is \(-\frac{4}{3}\))

However, you must know the specific slope of runway A to give a definitive answer.