Question

Find an equation for the perpendicular bisector of the line segment whose endpoints are left bracket, 3, comma, minus, 7, right bracket(3,−7) and left bracket, minus, 9, comma, minus, 1, right bracket(−9,−1).

Answer 1

The perpendicular bisector of the line segment with endpoints (3, -7) and (-9, -1) passes through the midpoint (-3, -4) and has a slope of 2, which is the negative reciprocal of the original line's slope. The equation of the perpendicular bisector is y = 2x + 2.

Step-by-step explanation:

To find the equation for the perpendicular bisector of the line segment with endpoints (3, -7) and (-9, -1), we first need to determine the midpoint of the line segment, which gives us the point through which the perpendicular bisector will pass. The midpoint is found by averaging the x-coordinates and the y-coordinates of the endpoints:

• Middle of x: (3 + (-9)) / 2 = -3
• Middle of y: (-7 + (-1)) / 2 = -4

So the midpoint is (-3, -4).

Next, we need to find the slope of the original line segment. The slope m is the rise over the run, calculated as:

• m = (y2 - y1) / (x2 - x1) = (-1 - (-7)) / (-9 - 3) = 6 / -12 = -1/2

The slope of the perpendicular bisector is the negative reciprocal of the original slope, so we get:

• m_perpendicular = -1 / (-1/2) = 2

Now we have a slope and a point. Using the point-slope form of a line (y - y1) = m(x - x1), the equation of the perpendicular bisector is:

• y - (-4) = 2(x - (-3))
• y + 4 = 2x + 6
• y = 2x + 6 - 4
• y = 2x + 2

Thus, the equation of the perpendicular bisector is y = 2x + 2.