The set of all points equidistant from two endpoints — built from the midpoint and a slope relationship.
The Mnemonic
Two Ingredients: Midpoint and Perpendicular Slope
The perpendicular bisector of a segment is the line that passes through the segment's midpoint at a 90° angle to it. Finding its equation combines two earlier tools directly: the Midpoint Formula gives the point the bisector must pass through, and the negative reciprocal of the original segment's slope gives the bisector's own slope.
The negative reciprocal means: flip the fraction, then flip the sign. If the original segment has slope 2/3, the perpendicular bisector has slope −3/2.
The perpendicular bisector crosses segment AB at its midpoint M, at a 90° angle — every point on this line is equidistant from A and B
💡 Memory Trick
"The set of all points equidistant from two endpoints." Every single point sitting on the perpendicular bisector is exactly the same distance from A as it is from B — that's the defining PROPERTY of this line, not just a side detail. The midpoint-plus-perpendicular-slope construction is simply the fastest way to write its equation.
Why It Works
Why Every Point on It Is Equidistant
Take any point P on the perpendicular bisector of segment AB, and connect it to both A and B. Since the bisector crosses AB at its midpoint M at exactly 90°, the two triangles PMA and PMB share the segment PM, have congruent legs (AM ≅ MB, from the definition of midpoint), and share a right angle at M — that's SAS, so △PMA ≅ △PMB. By CPCTC, PA ≅ PB — meaning P truly is equidistant from A and B.
This is exactly why perpendicular bisectors are the key construction behind the circumcenter (from Triangle Centers) — the circumcenter is equidistant from all three vertices specifically because it lies on the perpendicular bisector of every side.
Using It In A Proof
Building the Equation Step by Step
Finding a perpendicular bisector's equation is really a three-tool combination: Slope Formula, Midpoint Formula, and Slope-Intercept Form, used together in sequence.
1
Find the original segment's slope
Use the Slope Formula on the two given endpoints.
2
Find the negative reciprocal
Flip the fraction and switch the sign — this is the perpendicular bisector's slope.
3
Find the midpoint
Use the Midpoint Formula on the same two endpoints — this is the point the bisector passes through.
4
Write the equation
Substitute the perpendicular slope and the midpoint into y = mx + b, then solve for b as in Slope-Intercept Form.
Full Worked Example
Finding the Full Equation of a Perpendicular Bisector
Given: Segment endpoints A(2, 8) and B(6, 0). Find: the equation of its perpendicular bisector.
1
Find the slope of AB
m = (0−8)/(6−2) = −8/4 = −2.
2
Find the negative reciprocal
Flip −2 (or −2/1) to get −1/2, then switch the sign: the perpendicular slope is 1/2.
3
Find the midpoint of AB
M = ((2+6)/2, (8+0)/2) = (4, 4).
4
Write the equation using point-slope logic
Substituting into y = mx + b: 4 = (1/2)(4) + b, so 4 = 2 + b, so b = 2. Final equation: y = (1/2)x + 2.
Check: this line should pass through (4,4) — (1/2)(4)+2 = 2+2 = 4. ✓
🎯 Quick Worked Example
Segment CD has endpoints C(0, 0) and D(8, 4). Find the slope of its perpendicular bisector, and the midpoint it passes through.
1
Find the slope of CD. m = (4−0)/(8−0) = 4/8 = 1/2.
2
Find the negative reciprocal. Flip 1/2 to get 2/1, switch the sign: perpendicular slope = −2.
3
Find the midpoint. M = (4, 2).
📌 Exam Application
This topic is a favorite combination question specifically because it requires chaining THREE earlier formulas together (Slope, negative reciprocal, Midpoint) before even writing the final equation — practice recognizing this exact three-step chain, since it appears constantly in coordinate geometry problem sets.
⚠️ Most Common Perpendicular Bisector Mistakes
Trap 1 — Forgetting to flip the sign, only flipping the fraction (or vice versa): The negative reciprocal requires BOTH steps — flip the fraction AND change the sign. Doing only one of these gives a slope that's either parallel or has the wrong sign, not perpendicular.
Trap 2 — Using the original slope instead of the negative reciprocal when writing the final equation: After finding both the midpoint and the perpendicular slope, it's easy to accidentally substitute the ORIGINAL segment's slope back into the final equation instead of the new perpendicular one — always double check which slope value you're plugging in.
✓ Quick Self-Test
1) What two ingredients are needed to write a perpendicular bisector's equation? 2) Find the negative reciprocal of a slope of −4/5. 3) Why is every point on a perpendicular bisector equidistant from the two original endpoints? 4) Segment EF has endpoints E(1,1) and F(5,9) — find the midpoint and the perpendicular bisector's slope. 5) How does the perpendicular bisector connect to the circumcenter from Triangle Centers?