BananaLearnLearning LibraryGamesDistance between (2,1) and (8,9).
- Horizontal difference = 6; vertical difference = 8.
- d² = 36 + 64 = 100, so d = 10 units.
- 6 + 8 = 14 would measure a two-part grid route.
Turn the coordinate difference into a triangle
For A(1,2) and B(4,6), the horizontal displacement is 4−1=3 and the vertical displacement is 6−2=4. Travel across then up to form two perpendicular legs. The direct segment from A to B is the hypotenuse, so its length is √(3²+4²)=5.
A grid route has length 3+4=7, but the straight-line distance is 5. Distance in this formula means the direct Euclidean distance on perpendicular axes with a common scale; it does not measure a walking route constrained by roads.
| Points | Horizontal difference | Vertical difference | Distance |
|---|---|---|---|
| (1,2), (4,6) | 3 | 4 | 5 |
| (−2,3), (4,−5) | 6 | −8 | 10 |
| (3,−2), (3,5) | 0 | 7 | 7 |
| (0,0), (1,1) | 1 | 1 | √2 |
Derive the formula rather than memorising a sign pattern
For P(x₁,y₁) and Q(x₂,y₂), the two leg lengths are the absolute values of x₂−x₁ and y₂−y₁. Squaring removes their signs, so d²=(x₂−x₁)²+(y₂−y₁)². Distance is non-negative, so take the non-negative square root.
Keep each subtraction inside brackets: for −1 and 2, the difference is 2−(−1)=3. The expression (2−(−1))² is 9. Squaring the coordinates separately and then subtracting would calculate 4−1=3, a different expression.
Use squared distances when classifying a shape
To test whether two sides have equal length, compare their squared distances. If both squares are 25, both lengths are 5. This avoids unnecessary rounding. For a right triangle, test whether the largest squared side equals the sum of the other two squared sides.
For A(0,0), B(4,0), C(4,3), the squared lengths are AB²=16, BC²=9 and AC²=25. Since 16+9=25, the triangle is right-angled at B. The location of the right angle comes from the sides meeting there, not merely from the fact that a sum works.
A missing coordinate can have two solutions
Suppose (x,0) lies 5 units from (0,3). Then x²+(0−3)²=25, so x²=16 and x=4 or x=−4. Both points satisfy the distance. Taking only a positive x loses the point on the other side of the vertical axis.
Distance itself is non-negative; a coordinate may be negative. This is why the final distance uses the positive square root while solving a coordinate equation can require both signs. Check both candidates on a quick sketch.
Know when to leave a surd
For (1,1) to (3,4), distance is √13 units. Unless a decimal is requested, √13 is exact and clearer than an early rounded approximation. If comparing lengths, compare 13 with the other squared distance before rounding.
The distance formula does not find a midpoint. The midpoint averages each coordinate; distance uses differences and squares. Writing down what the question asks before choosing a formula prevents this common mix-up.
Worked examples: follow each decision
Distance between (−1,−2) and (2,2).
- Differences: 2 − (−1) = 3 and 2 − (−2) = 4.
- d = √(9 + 16) = 5 units.
- Brackets keep subtraction of negative coordinates clear.
Show that A(1,1), B(5,1), C(5,4) form a right triangle.
- AB²=16; BC²=9; AC²=25.
- 16+9=25, so the two shorter sides meet at a right angle.
- They meet at B. The check identifies the right angle and the lengths 4, 3 and 5.
13 questions with explained answers
Attempt a short group of questions before opening the solutions. Use a drawing or written method where helpful. Explain the answer, then try a similar question with a detail changed.
(0,0) to (6,8)
Show answer and reasoning
10 units
√(36 + 64) = 10.
(3,−2) to (3,5)
Show answer and reasoning
7 units
Vertical separation is 5 − (−2).
(−4,1) to (2,1)
Show answer and reasoning
6 units
The y-coordinates are equal.
(0,0) to (1,1)
Show answer and reasoning
√2 units
Exact distance is √(1 + 1).
Does reversing the points change distance?
Show answer and reasoning
No
The differences change sign but their squares do not.
(−3,−4) to (0,0)
Show answer and reasoning
5 units
√(9+16)=5.
(1,1) to (3,4)
Show answer and reasoning
√13 units
Differences 2 and 3 give 4+9.
(−2,5) to (4,5)
Show answer and reasoning
6 units
Horizontal separation only.
Points (x,0) and (0,3) are 5 units apart. Find x.
Show answer and reasoning
4 or −4
x²+9=25.
A(0,0), B(3,0), C(0,4): longest side?
Show answer and reasoning
BC = 5
AB=3, AC=4 and BC²=9+16.
Can distance be −7?
Show answer and reasoning
No
Distance is a non-negative length.
Why is (5−2)² not 5²−2²?
Show answer and reasoning
They are 9 and 21 respectively
The brackets require subtraction before squaring.
All coordinates double. What happens to distances?
Show answer and reasoning
They double
Both differences double; their squared sum quadruples and its root doubles.
Check what you can now explain
- Have you subtracted matching coordinates inside brackets?
- Is the question about a direct distance or a route?
- Could a missing coordinate have two solutions?
Keep these distinctions clear
- Square the whole difference: (4 − 1)² = 9, not 4² − 1² = 15.
- The formula assumes perpendicular axes with the same unit scale.
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Keep the ideas connected
- Grade 8 Pythagoras’ Theorem and the Right-Triangle Test
- Linear equations in two variables for Grade 9
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