What is the equation of the perpendicular bisector of the line segment connecting the points (2, 8) and (6, 4)?

Correct — D. The perpendicular bisector passes through the midpoint and meets the segment at 90°.

Midpoint: ((2 + 6)⁄2, (8 + 4)⁄2) = (4, 6)
Slope of the segment: (4 − 8)⁄(6 − 2) = −4⁄4 = −1
Perpendicular slope, the negative reciprocal of −1: 1

Point-slope form through (4, 6): y − 6 = 1 × (x − 4)
Add 6 to both sides: y = x + 2 → option (d)

Why the others are wrong

• (a)
  y=−x−10 — y = −x − 10 has slope −1, the segment's own slope, so it runs parallel to the segment. It also misses the midpoint: at x = 4 it gives y = −14.
• (b)
  y=x−2 — y = x − 2 has the right slope, 1, but misses the midpoint: at x = 4 it gives y = 2, not 6. It is perpendicular to the segment without bisecting it.
• (c)
  y=−x+10 — y = −x + 10 is the line through the two given points: (2, 8) and (6, 4) both satisfy it. It is the segment's own line, not the line perpendicular to it.

The perpendicular bisector of a segment is the set of points equidistant from its two ends. To write its equation you need one point and a slope.

The point is the midpoint. The slope is the negative reciprocal of the segment's slope, because two perpendicular lines, neither vertical, have slopes whose product is −1.

Key facts

• Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂)⁄2, (y₁ + y₂)⁄2).
• Slope through two points = (y₂ − y₁)⁄(x₂ − x₁).
• Perpendicular lines, neither vertical, have slopes whose product is −1.
• Point-slope form: y − y₁ = m(x − x₁).

Common traps

• Using the segment's slope (−1) instead of its negative reciprocal (1).
• Stopping at the slope: y = x − 2 and y = x + 2 both have slope 1, and only the midpoint (4, 6) separates them.

No directly related past PYQ was found.