7 min readConcept sheet1-mark MCQ, 2-mark and 3-mark questions

Distance formula

The distance between P(x₁, y₁) and Q(x₂, y₂) is PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes from Pythagoras' theorem.

Distance formulaPQ = √[(x₂ − x₁)² + (y₂ − y₁)²]

Distance from the originOP = √(x² + y²)

Order does not matter(x₁ − x₂)² = (x₂ − x₁)², so either point can come first

The answer isNever negative; give it in units and simplify any surd

How to draw the figure and derive the formula

1
Plot P(x₁, y₁) and Q(x₂, y₂). Draw a horizontal line through P and a vertical line through Q. They meet at R(x₂, y₁).
2
PR is horizontal, so PR = x₂ − x₁. QR is vertical, so QR = y₂ − y₁.
3★
∠PRQ = 90°. By Pythagoras' theorem, PQ² = PR² + QR².
you know Pythagoras' theorem from earlier classes; here you only use it, you do not prove it
4
So PQ² = (x₂ − x₁)² + (y₂ − y₁)², and PQ = √[(x₂ − x₁)² + (y₂ − y₁)²].

★ key step

Example 1: distance between two points

Problem

Find the distance between A(−2, 3) and B(4, −5).

Working

x₁ = −2, y₁ = 3, x₂ = 4, y₂ = −5
Formula: AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
AB = √[(4 − (−2))² + (−5 − 3)²]
AB = √[6² + (−8)²] = √(36 + 64)
AB = √100 = 10 units

Answer

AB = 10 units

Example 2: an unknown coordinate

Problem

The distance between P(x, 4) and Q(2, −2) is 10 units. Find the possible values of x.

Working

Formula: PQ² = (x₂ − x₁)² + (y₂ − y₁)²
10² = (x − 2)² + (4 − (−2))²
100 = (x − 2)² + 36
(x − 2)² = 64
x − 2 = 8 or x − 2 = −8
x = 10 or x = −6

Answer

x = 10 or x = −6

Example 3: a water tank on the road

Problem

On a village map, the x-axis is a straight road. Two houses are at A(−1, 4) and B(5, 2). A water tank is to be built on the road at the same distance from both houses. Where should it go?

Working

A point on the x-axis has y = 0. Let the tank be at P(x, 0).
PA = PB, so PA² = PB².
(x + 1)² + (0 − 4)² = (x − 5)² + (0 − 2)²
x² + 2x + 1 + 16 = x² − 10x + 25 + 4
2x + 17 = −10x + 29
12x = 12, so x = 1
Check: PA² = 2² + 4² = 20 and PB² = 4² + 2² = 20.

Answer

The tank should be at P(1, 0).

Common mistakes

Squaring a negative: (−8)² = 64, not −64.
Writing √(36 + 64) = 6 + 8. Add first, then take the root: √100 = 10.
Losing an answer: (x − 2)² = 64 gives x − 2 = 8 or −8.
When two distances are equal, square both sides (PA² = PB²). This removes the square roots.

In real life

On a town map drawn on a grid, this formula gives the straight-line distance from your home to school. The road route is usually longer.

Exam angle

1 marks: “The distance of the point (−5, 12) from the origin is:” √(25 + 144) = √169 = 13 units.

2 marks: “Find the values of y if the distance between (2, y) and (−4, 3) is 10 units.” 6² + (y − 3)² = 100, so (y − 3)² = 64 and y − 3 = ±8. y = 11 or y = −5.

3 marks: “A point P(x, y) is at the same distance from A(2, 1) as from B(−4, 3). Write an equation in x and y that P must satisfy.” PA² = PB²: (x − 2)² + (y − 1)² = (x + 4)² + (y − 3)². Expand and simplify: −12x + 4y − 20 = 0, that is 3x − y + 5 = 0.

Remember it

Difference, square, add, root: (x-gap)² + (y-gap)², then take √.