A line L is the perpendicular bisector of the line segment connecting points A(2, 5) and B(8, −1). What is the y-intercept of line L?

Correct — A.
Midpoint of AB = ((2 + 8) ⁄ 2, (5 − 1) ⁄ 2) = (5, 2)
Slope of AB = (−1 − 5) ÷ (8 − 2) = −6 ÷ 6 = −1
Slope of L = negative reciprocal of −1 = 1

Line through (5, 2) with slope 1: y − 2 = 1 × (x − 5)
Simplify: y = x − 3
Put x = 0: y-intercept = −3 → option (a)

Why the others are wrong

• (b)
  2 — 2 is the y-coordinate of the midpoint (5, 2), where L crosses AB. The y-intercept is where L crosses the y-axis, at x = 0, and there y = −3.
• (c)
  3 — 3 has the right size and the wrong sign. The line y = x − 3 meets the y-axis at y = −3, below the origin, not above it.
• (d)
  5 — 5 is the midpoint's x-coordinate, not a point on the y-axis. Putting x = 0 into y = x − 3 gives −3.

A perpendicular bisector does two jobs: it passes through the midpoint of the segment and crosses it at 90°.

The midpoint gives you a point on the line. The slope comes from the segment: two perpendicular lines, neither vertical, have slopes that multiply to −1, so L's slope is the negative reciprocal of AB's.

With a point and a slope, write y − y₁ = m(x − x₁), then put x = 0 to read off the y-intercept.

Key facts

• Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂) ⁄ 2, (y₁ + y₂) ⁄ 2).
• Two perpendicular lines, neither vertical, have slopes whose product is −1.
• Every point on the perpendicular bisector of AB is equally far from A and B.
• The y-intercept of a line is its y-value at x = 0.

Common traps

• Giving L the slope of AB (−1) instead of its negative reciprocal, which gives y = −x + 7 and an intercept of 7
• Slipping on the sign in the last step: y = x − 5 + 2 is x − 3, not x + 3

No directly related past PYQ was found.