Topic 3.5 · Applications and Interpretation

Every point on it is the same distance from both

Two water pumps sit at A(1, 2) and B(7, 6) on a map of a village, with distances in kilometres. Where is the line of houses that are exactly as far from one pump as the other? That line is the perpendicular bisector, and that sentence is its definition, not a fact about it.

at the midpoint
3.61distance to A
3.61distance to B
equalverdict

The two distances stay equal all the way along, which is what the line is for. On the fourth view they do not.

Three steps, in this order

For A(1, 2) and B(7, 6):

  1. Midpoint. Average the coordinates: ((1+7)/2, (2+6)/2) = (4, 4). The bisector goes through it.
  2. Gradient of AB. (6 − 2)/(7 − 1) = 4/6 = 2/3.
  3. Negative reciprocal. Flip it and change the sign: −3/2. That is the gradient of the bisector.

Then put the point and the gradient into a line:

  1. y − 4 = −1.5(x − 4)
  2. y = −1.5x + 10, or 3x + 2y = 20 if you prefer it tidy

Negative reciprocal means two things, and people do one. Turn 2/3 upside down to get 3/2, and put a minus on it. A gradient of 3/2 or of −2/3 is half a job. The test is that the two gradients multiply to −1: 2/3 × −3/2 = −1.

Why the wrong line still looks right

Use 2/3 instead of −3/2 and you still get a line through the midpoint, still at a believable angle, still with a tidy equation. Nothing about it looks wrong. But at x = 0 it sits at y = 1.33, and from there it is 1.20 km to pump A and 8.41 km to pump B. One pump is seven times as far as the other. That is not a boundary between them.

So the check is free, and it is the definition. Take any point on your line, away from the midpoint, and measure to both:

  1. At (0, 10): to A is √(1 + 64) = √65 = 8.06, to B is √(49 + 16) = √65 = 8.06
  2. At (6, 1): to A is √(25 + 1) = √26 = 5.10, to B is √(1 + 25) = √26 = 5.10

The two square roots come out with the same numbers underneath in a different order. That is what being on the bisector looks like in the arithmetic.

A second one, with negatives in it

P(−2, 5) and Q(4, 1):

  1. Midpoint: (1, 3)
  2. Gradient of PQ: (1 − 5)/(4 + 2) = −4/6 = −2/3
  3. Negative reciprocal: 3/2. A negative gradient has a positive perpendicular, which catches people who expect a minus every time.
  4. y − 3 = 1.5(x − 1), so y = 1.5x + 1.5

Check at (3, 6): to P is √(25 + 1) = 5.10, to Q is √(1 + 25) = 5.10. Equal, so it is right.

Given a segment and its midpoint instead

The syllabus also sets this the other way round: you are given the line segment, or its equation, and its midpoint. Then you already have step 1 and step 2, and only the negative reciprocal is left. If the segment is given as an equation like y = 2x + 1, its gradient is 2 and the bisector's is −½; put that through the midpoint you were given.

This is the whole of Voronoi. In 3.6 a Voronoi boundary between two sites is the perpendicular bisector of the two sites, for exactly this reason: on it you are equally close to both, so it is where the nearest one changes. If this page is solid, that one is mostly bookkeeping.

On the GDC: checking a bisector

There is no perpendicular-bisector command. What the machine is for here is the check: two distances, typed in as they stand, which takes about ten seconds and catches the negative-reciprocal error every time.

When you may use it. Applications. A calculator is allowed in every paper.

TI-Nspire CX II

  1. In a Calculator page, for the point (0, 10) against A(1, 2): ctrl x² for the root, then (0-1) x² +(10-2) x²: 8.0623. The x² key, not ^2, which opens a superscript box.
  2. Against B(7, 6), the same way: 8.0623
  3. Equal, so the point is on the bisector. If they differ, the gradient is wrong.
  4. To see it: ctrl doc → Add Graphs, plot f1(x)=-1.5x+10, then menu → Geometry → Points & Lines → Point to drop A and B on and look.

Casio fx-CG50

  1. MENU → Run-Matrix, then √((0-1)²+(10-2)²) using SHIFT x² for the root: 8.0623
  2. Then √((0-7)²+(10-6)²): 8.0623
  3. To see it: MENU → Graph, enter Y1=-1.5X+10 and F6 DRAW
  4. SHIFT F3 V-Window and set equal scales, or a perpendicular line will not look perpendicular

The mark people lose. Half a negative reciprocal. From 2/3, writing 3/2 or −2/3 instead of −3/2. Both are wrong and both look like you did something. Multiply your two gradients together before you write the equation down: if the answer is not −1, stop. And on the screen, a bisector only looks perpendicular when the x and y scales match, so do not judge it by eye on a default window.

Your turn

1. Find the gradient of the perpendicular bisector of A(1, 2) and B(7, 6). Give it as a decimal.

2. What is the y-intercept of that bisector? Give it to 1 decimal place.

3. For P(−2, 5) and Q(4, 1), find the y-intercept of the perpendicular bisector, to 1 decimal place.

4. A student gives the bisector of A(1, 2) and B(7, 6) as y = (2/3)x + 4/3. It does pass through the midpoint. How would you show them it is wrong in one line?

Question 4. A student gives the bisector of A(1, 2) and B(7, 6) as y equals two thirds x plus four thirds. It does pass through the midpoint. How would you show them it is wrong in one line?
Where the marks go

1 markThe midpoint.

1 markThe gradient of the segment.

1 markThe negative reciprocal.

1 markA correct equation of the line.

Four marks for three short calculations and a substitution, which makes this one of the best-paid sub-topics on the course. Show all three numbers even if you can see the answer, because each one is a mark.

Want a verdict on your own draft?

These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.

Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.

Send me your question, free

Already have a full draft? Have the whole thing reviewed against all five criteria, $99.