Question

Find the equation of the line perpendicular bisector of the line segment joining: (-3,-4) and (5,-6).

a) y=x+1
b) y=−x−1
c) y=2x−1
d) y=−2x+1

Answer 1

To find the equation of the line perpendicular bisector of the line segment joining (-3,-4) and (5,-6), we can find the midpoint, the slope of the line segment, the slope of the perpendicular bisector, and then use the point-slope form to find the equation. The equation is y = 4x - 9.

Step-by-step explanation:

To find the equation of the line perpendicular bisector of the line segment, we first need to find the midpoint of the line segment. The midpoint is the average of the x-coordinates and the average of the y-coordinates of the two given points. The midpoint of the line segment joining (-3,-4) and (5,-6) is ((-3+5)/2, (-4-6)/2), which simplifies to (1, -5).

Next, we need to find the slope of the line segment. The slope of a line segment is given by the formula: slope = (y2 - y1) / (x2 - x1). Using the coordinates of the two given points, the slope of the line segment joining (-3,-4) and (5,-6) is (-6 - (-4)) / (5 - (-3)), which simplifies to -2 / 8, or -1/4.

Finally, the slope of the perpendicular bisector of a line segment is the negative reciprocal of the slope of the line segment. So, the slope of the perpendicular bisector is 4.

Now that we have the midpoint and the slope of the perpendicular bisector, we can use the point-slope form of a line to find the equation. The point-slope form is: y - y1 = m(x - x1), where (x1, y1) is the midpoint and m is the slope. Substituting the values, we get y - (-5) = 4(x - 1), which simplifies to y + 5 = 4x - 4. Rearranging the equation, we get y = 4x - 9.

Therefore, the equation of the line perpendicular bisector of the line segment joining (-3,-4) and (5,-6) is y = 4x - 9.

Answer 2

The equation of the line perpendicular bisector of the line segment joining: (-3,-4) and (5,-6) is y = -x - 1

Therefore,correct answer is b) y = -x - 1

Step-by-step explanation:

The equation of the perpendicular bisector of a line segment joining two points
`\((x_1, y_1)\)` and
`\((x_2, y_2)\)` can be found using the midpoint formula and the negative reciprocal of the slope of the line segment. The midpoint M is given by
`\(\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\)`, and the slope of the line segment m is calculated as
`\(\frac{{y_2 - y_1}}{{x_2 - x_1}}\)`.

The negative reciprocal of m is used to find the slope of the perpendicular bisector. Using the midpoint and the new slope, the equation of the line is determined. In this case, the result is y = -x - 1.

Understanding the properties of perpendicular bisectors is fundamental in analytical geometry. The negative reciprocal of the slope ensures that the lines are perpendicular, and the midpoint formula is used to find the center of the line segment. This process is crucial in various applications, such as geometry and engineering.

Therefore,correct answer is b) y = -x - 1