Chapter 4 · Geometry and Equations · Concept 1
4-3 Distance Between Two Points
There is no new idea in this lesson. There is a right-angled triangle you have known for years, hiding between any two points on a grid — and once you see it, the formula stops being something to remember.
Guiding question
How far apart are two points, when all you have is their addresses?
Learning outcome. Calculate the distance between two points.
Pythagoras, with the sides worked out by subtracting
for A(x₁, y₁) and B(x₂, y₂)
AB = √((x₂ − x₁)2 + (y₂ − y₁)2)
Where it comes from — one picture.
Go across from A, then up to B. Those two moves are the legs of a right-angled triangle, and AB is its longest side. The across-leg is x₂ − x₁ and the up-leg is y₂ − y₁, so a² + b² = c² gives exactly the formula above. There is nothing else in it.
And when the distance is the thing you are TOLD:
square both sides
That gets rid of the root and leaves an ordinary equation. It will be a quadratic, so expect two answers — and both are usually correct, because there are genuinely two points at that distance.
Every one of these is the same picture with different numbers. Watch where the two legs come from.
across: 5 − 2 = 3 · up: 10 − 6 = 4
AB² = …
2. The distance between A(3, −2) and B(x, 1) is 5. Find x.
- 1
Write the distance out with the letter still in it.
√((x − 3)2 + (1 − (−2))2) = 5Keep the brackets round −2. That subtraction is 1 + 2 = 3, and losing it is the usual slip.
Work with a partner. For question 2, once you both have your two answers, plot them. Seeing two points sitting the same distance away on either side is what makes the ± stop feeling like a rule.
1. Find the distance.
1. Find the distance.
2–4. Find the missing coordinate.
The distance between the points (x, 7) and (3x − 1, −5) is 13. Find every value of x.
Squaring a surd distance wrongly.
(5√5)² is 25 × 5 = 125, not 25. Both parts get squared.
Dropping the ± after rooting.
(x − 3)² = 16 gives x − 3 = ±4, so two answers. There really are two points at that distance.
Losing a sign in a subtraction.
1 − (−2) is 3, not −1. Keep the brackets round every negative coordinate.
Rooting each term separately.
It is √(a² + b²), not √a² + √b². The root goes over the whole sum, because it is one triangle.
Simplifying the legs too late.
When the letter appears in both points, work out the leg first — (3x − 1) − x is 2x − 1, and expanding before that is a mess.
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