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First find the mid-point of the line segment which will be the point of intersection of the perpendicular bisector. Then find the slope or gradient of the line segment whose negative reciprocal will be the perpendicular bisector's slope or gradient. Then use y -y1 = m(x -x1) to find the equation of the perpendicular bisector.

Mid-point: (7p+p)/2 and (3q+q)/2 = (4p, 2q)

Slope or gradient: 3q-q/7p-p = 2q/6p = q/3p

Slope of perpendicular bisector: -3p/q

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

y = -3px/q+12p2/q+2q

Multiply all terms by q to eliminate the fractions:

qy = -3px+12p2+2q2

Which can be expressed in the form of: 3px+qy-12p2-2q2 = 0

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Q: How do you work out and find the perpendicular bisector equation meeting the straight line segment of p q and 7p 3q?
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