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A is the point (6,1) and B is the point (2,7). Find the equation of the perpendicular bisector of AB. Give your answer in the form y=mx+c.

A is the point (6,1)(6,1) and B is the point (2,7)(2,7). Find the equation of the perpendicular bisector of AB. Give your answer in the form y=mx+c.y=mx+c.

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Q. A is the point (6,1)(6,1) and B is the point (2,7)(2,7). Find the equation of the perpendicular bisector of AB. Give your answer in the form y=mx+c.y=mx+c.
  1. Find Midpoint of AB: Find the midpoint of AB to determine where the perpendicular bisector will cross AB. Midpoint formula is (x1+x22,y1+y22) \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) . Midpoint = (6+22,1+72)=(82,82)=(4,4) \left( \frac{6 + 2}{2}, \frac{1 + 7}{2} \right) = \left(\frac{8}{2}, \frac{8}{2}\right) = (4, 4) .
  2. Calculate Slope of AB: Calculate the slope of AB using the formula (y2−y1)/(x2−x1)(y_2 - y_1) / (x_2 - x_1).\newlineSlope of AB = (7−1)/(2−6)=6/−4=−3/2(7 - 1) / (2 - 6) = 6 / -4 = -3/2.
  3. Find Slope of Perpendicular Bisector: Find the slope of the perpendicular bisector. The slope of the perpendicular line is the negative reciprocal of the slope of ABAB.\newlineSlope of the perpendicular bisector = −1/(−32)=23-1 / (-\frac{3}{2}) = \frac{2}{3}.
  4. Write Equation in Point-Slope Form: Use the point-slope form to write the equation of the perpendicular bisector. Point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line.\newlineUsing the midpoint (4,4)(4, 4) and the slope 23\frac{2}{3}, the equation is y−4=(23)(x−4)y - 4 = \left(\frac{2}{3}\right)(x - 4).
  5. Convert to Slope-Intercept Form: Simplify the equation to slope-intercept form y=mx+cy = mx + c.y−4=(23)x−(23)⋅4y - 4 = \left(\frac{2}{3}\right)x - \left(\frac{2}{3}\right)\cdot4y=(23)x−(83)+4y = \left(\frac{2}{3}\right)x - \left(\frac{8}{3}\right) + 4y=(23)x−(83)+(123)y = \left(\frac{2}{3}\right)x - \left(\frac{8}{3}\right) + \left(\frac{12}{3}\right)y=(23)x+(43)y = \left(\frac{2}{3}\right)x + \left(\frac{4}{3}\right)

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