Final answer:
To find a line perpendicular to the one through (-4, 2) and (4, -4), we calculate the original line's slope, which is -3/4, and then look for a line with a slope of 4/3 (the negative reciprocal). Option c) has the needed slope, making it the correct choice.
Step-by-step explanation:
To determine which line is perpendicular to the one passing through the points (-4, 2) and (4, -4), first calculate the slope of this line. The slope formula is (y2 - y1) / (x2 - x1). Substituting the points, we get (-4 - 2) / (4 - (-4)) = -6 / 8, which simplifies to -3/4. A line perpendicular to this would have a slope that is the negative reciprocal, so it would be 4/3.
Now, look at the equations given and put them into slope-intercept form (y = mx + b) if necessary, to identify their slopes:
- a) 3x + 4y = 16 => 4y = -3x + 16 => y = (-3/4)x + 4 (not perpendicular)
- b) 3x - 4y = 4 => 4y = 3x - 4 => y = (3/4)x - 1 (not perpendicular)
- c) 4x + 3y = 15 => 3y = -4x + 15 => y = (-4/3)x + 5 (perpendicular)
- d) 4x - 3y = 9 => 3y = 4x - 9 => y = (4/3)x - 3 (not perpendicular)
Therefore, the correct answer is option c), as its slope is 4/3, which is the negative reciprocal of the original line's slope.