To find the equations of runways B and C in relation to runway A, we first need to determine the slope of runway A using the given points \((-4, 0)\) and \((0, 3)\).
- Calculating the slope of runway A: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{0 - (-4)} = \frac{3}{4} \]
The slope of runway A is \(\frac{3}{4}\).
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Finding the equation for runway B (parallel to A): The slope of runway B will be the same as runway A, which is \(\frac{3}{4}\). We can use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) = (0, -4) \) and \( m = \frac{3}{4} \).
Plugging in these values: \[ y - (-4) = \frac{3}{4}(x - 0) \] \[ y + 4 = \frac{3}{4}x \] \[ y = \frac{3}{4}x - 4 \]
Therefore, the equation for runway B is: \[ y = \frac{3}{4}x - 4 \]
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Finding the equation for runway C (perpendicular to A): The slope of a line perpendicular to runway A is the negative reciprocal of \(\frac{3}{4}\). Thus, the slope for runway C is: \[ m = -\frac{4}{3} \]
Again using point-slope form with the point \((0, -4)\): \[ y - (-4) = -\frac{4}{3}(x - 0) \] \[ y + 4 = -\frac{4}{3}x \] \[ y = -\frac{4}{3}x - 4 \]
Therefore, the equation for runway C is: \[ y = -\frac{4}{3}x - 4 \]
Now, to summarize the equations we found:
- Runway B: \(y = \frac{3}{4}x - 4\)
- Runway C: \(y = -\frac{4}{3}x - 4\)
Now, selecting the correct choices from the given options:
- A: \(y = -\frac{3}{4}x - 4\) (Not correct)
- B: \(y = \frac{4}{3}x - 4\) (Not correct)
- C: \(y = -\frac{4}{3}x - 4\) (Correct, corresponds to Runway C)
- D: \(y = \frac{3}{4}x - 4\) (Correct, corresponds to Runway B)
Thus, the correct options are:
- C: \(y = -\frac{4}{3}x - 4\)
- D: \(y = \frac{3}{4}x - 4\)
2 answers