Question

Write an equation of the line that is the perpendicular bisector of the segment with endpoints a (4,1) and b (8,3)

Answer 1

To find the equation of the line that is the perpendicular bisector of the segment with endpoints (4,1) and (8,3), we can calculate the midpoint of the segment and then determine the slope of the perpendicular bisector. Using the midpoint (6, 2) and a slope of -2, we can write the equation of the perpendicular bisector as y = -2x + 14.

Step-by-step explanation:

To find the equation of the line that is the perpendicular bisector of the segment with endpoints a (4,1) and b (8,3), we can first find the midpoint of the segment. The midpoint formula is given by:

x = (x1 + x2) / 2

y = (y1 + y2) / 2

Using the given endpoints, we can calculate the midpoint:

x = (4 + 8) / 2 = 6

y = (1 + 3) / 2 = 2

So, the midpoint of the segment is (6, 2).

The slope of the original segment can be calculated using the formula:

slope = (y2 - y1) / (x2 - x1)

slope = (3 - 1) / (8 - 4) = 2 / 4 = 0.5

Since the perpendicular bisector of a segment has a slope that is the negative reciprocal of the original segment's slope, the slope of the perpendicular bisector is -1 / 0.5 = -2.

Using the point-slope form of a line, where the equation is y - y1 = m(x - x1), we can write the equation of the perpendicular bisector using the midpoint (6, 2) and the slope -2:

y - 2 = -2(x - 6)

y - 2 = -2x + 12

y = -2x + 14

So, the equation of the line that is the perpendicular bisector of the segment with endpoints a (4,1) and b (8,3) is y = -2x + 14.

Answer 2

To find the equation of the perpendicular bisector of the segment with endpoints A (4,1) and B (8,3), calculate the midpoint, find the negative reciprocal of the slope of AB, and use the point-slope form with these values to get y = -2x + 14.

Step-by-step explanation:

The equation of a line that is the perpendicular bisector of a segment with endpoints A (4,1) and B (8,3) can be found following a series of steps. First, calculate the midpoint of the segment which will be a point on the bisector. Then find the slope of the line segment AB, and the slope of the perpendicular bisector will be the negative reciprocal of this slope. Finally, use the point-slope form to write the equation of the perpendicular bisector.

To calculate the midpoint (M), use the formula: M = ((x1 + x2)/2, (y1 + y2)/2). For A (4,1) and B (8,3), the midpoint is (6,2).

The slope of AB is (y2 - y1)/(x2 - x1) = (3 - 1)/(8 - 4) = 0.5. The slope of the perpendicular bisector (m') is the negative reciprocal, so m' = -2.

The equation of the line with slope -2 passing through the midpoint (6,2) is obtained using the point-slope form: y - y1 = m'(x - x1). Substituting the values, we get y - 2 = -2(x - 6), which simplifies to y = -2x + 14 as the final equation of the perpendicular bisector.