Best Answer

The perpendicular bisector of a line has a gradient (m') which is the negative reciprocal of the gradient (m) of the line, and passes through the mid-point of the line.

The equation of a line with gradient m through a point (x0, y0) has an equation of the form:

y - y0 = m(x - x0)

The gradient m of a line between two points (x0, y0) and (x1, y1) is given by:

m = change_in_y / change_in_x = (y1 - y0) / (x1 - x0)

Thus the line through (p, q) and (7p, 3q) has gradient:

m = (3q - q) / (7p - p) = 2q / 6p = q/3p

and the perpendicular bisector has gradient:

m' = -1 / m = -1 / (q/3p) = -3p/q

The midpoint of the line through (p, q) and (3p, 3q) is:

midpoint = ((p + 7p)/2), (q + 3q)/2) = (4p, 2q)

Thus the perpendicular bisector of the line between (p, q) and (7p, 3q) has equation:

y - 2q = -3p/q (x - 4p)

â†’ qy - 2qÂ² = -3px + 12pÂ²

â†’ qy + 3px = 12pÂ² + 2qÂ²

Additional Information:-

Final answer in its general form: 3px+qy-12p^2-2q^2 = 0

Q: What is the perpendicular bisector equation of the line joined by the points of p q and 7p 3q showing work and final answer in its general form?

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Points: (7, 3) and (-6, 1) Midpoint: (0.5, 2) Slope: 2/13 Perpendicular slope: -13/2 Perpendicular bisector equation: y-2 = -13/2(x-0.5) => 2y = -13x+10.5

The equation will be a perpendicular bisector equation of the given points:- Points: (-1, 3) and (-2, -5) Midpoint: (-3/2, -1) Slope: 8 Perpendicular slope: -1/8 Perpendicular equation: y--1 = -1/8(x--3/2) => y = -1/8x-3/16-1 Therefore the perpendicular bisector equation is: y = -1/8x -19/16

Points: (s, 2s) and (3s, 8s) Slope: (8s-2s)/(3s-s) = 6s/2s = 3 Perpendicular slope: -1/3 Midpoint: (s+3s)/2 and (2s+8s)/2 = (2s, 5s) Equation: y-5s = -1/3(x-2s) => 3y-15s = -1(x-2s) => 3y = -x+17x Perpendicular bisector equation in its general form: x+3y-17s = 0

Points: (2, 3) and (5, 7) Length of line: 5 Slope: 4/3 Perpendicular slope: -3/4 Midpoint: (3.5, 5) Bisector equation: 4y = -3x+30.5 or as 3x+4y-30.5 = 0

Related questions

Points: (7, 3) and (-6, 1) Midpoint: (0.5, 2) Slope: 2/13 Perpendicular slope: -13/2 Perpendicular equation: y-2 = -13/2(x-0.5) => 2y = -13x+10.5 Perpendicular bisector equation in its general form: 26x+4y-21 = 0

Points: (s, 2s) and (3s, 8s) Midpoint: (2s, 5s) Slope: 3 Perpendicular slope: -1/3 Perpendicular bisector equation: y-5s = -1/3(x-2s) => 3y = -x+17s In its general form: x+3y-17s = 0

8

Points: (-2, 2) and (6, 4) Midpoint: (2, 3) Slope: 1/4 Perpendicular slope: -4 Perpendicular bisector equation: y-3 = -4(x-2) => y = -4x+11

Points: (7, 3) and (-6, 1) Midpoint: (0.5, 2) Slope: 2/13 Perpendicular slope: -13/2 Perpendicular bisector equation: y-2 = -13/2(x-0.5) => 2y = -13x+10.5

The equation will be a perpendicular bisector equation of the given points:- Points: (-1, 3) and (-2, -5) Midpoint: (-3/2, -1) Slope: 8 Perpendicular slope: -1/8 Perpendicular equation: y--1 = -1/8(x--3/2) => y = -1/8x-3/16-1 Therefore the perpendicular bisector equation is: y = -1/8x -19/16

Points: (s, 2s) and (3s, 8s) Slope: (8s-2s)/(3s-s) = 6s/2s = 3 Perpendicular slope: -1/3 Midpoint: (s+3s)/2 and (2s+8s)/2 = (2s, 5s) Equation: y-5s = -1/3(x-2s) => 3y-15s = -1(x-2s) => 3y = -x+17x Perpendicular bisector equation in its general form: x+3y-17s = 0

Points: (h, k) and (3h, -5k) Slope: -3k/h Perpendicular slope: h/3k Midpoint: (2h, -2k) Perpendicular equation: y--2k = h/3k(x-2h) Multiply all terms by 3k: 3ky--6k2 = h(x-2h) Equation in terms of 3ky = hx-2h2-6k2 Perpendicular bisector equation in its general form: hx-3ky-2h2-6k2 = 0

8

Points: (2, 3) and (5, 7) Length of line: 5 Slope: 4/3 Perpendicular slope: -3/4 Midpoint: (3.5, 5) Bisector equation: 4y = -3x+30.5 or as 3x+4y-30.5 = 0

Points: (-4, 8) and (0, -2) Slope: (8--2)/((-4-0) = -5/2 Perpendicular slope: 2/5 Midpoint: (-4+0)/2, (8-2)/2 = (-2, 3) Equation: y-3 = 2/5(x--2) Multiply all terms by 5: 5y-15 = 2(x--2) => 5y = 2x+19 Perpendicular bisector equation in its general form: 2x-5y+19 = 0

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