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Pythagorean Theorem:
Distance

Learn how the Pythagorean Theorem connects to the coordinate plane and discover how to find the distance between any two points.

Pre-Algebra

What is Distance?

In everyday life, distance is the length of the shortest path between two locations. On a coordinate plane, it is the length of the straight line connecting two points.

\(x\)
\(y\)
\(A\)
\(B\)
distance = ?

The Right Triangle Connection

When you draw a straight line between two points on a coordinate plane, you can always form a right triangle underneath it. The horizontal and vertical legs are easy to count, and the slanted line is the hypotenuse.

\(A\)
\(B\)
\(C\)
horizontal leg
vertical leg
hypotenuse = distance

Reviewing \( a^2 + b^2 = c^2 \)

The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.

\(a\)
\(b\)
\(c\)

The Distance Formula

When we apply the Pythagorean Theorem to two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we get the distance formula:

\( (x_2 - x_1) \)
The horizontal distance between the two points (leg \( a \))
\( (y_2 - y_1) \)
The vertical distance between the two points (leg \( b \))
\( d \)
The straight-line distance (hypotenuse \( c \))

Where the Formula Comes From

1
Plot two points on the coordinate plane: \( (x_1, y_1) \) and \( (x_2, y_2) \).
2
Draw a right triangle using the horizontal and vertical differences as legs.
3
Horizontal leg = \( |x_2 - x_1| \), Vertical leg = \( |y_2 - y_1| \).
4
Apply \( a^2 + b^2 = c^2 \): \( (x_2 - x_1)^2 + (y_2 - y_1)^2 = d^2 \).
5
Solve for \( d \) by taking the square root of both sides.

Distance vs. Horizontal/Vertical Counting

A common confusion: "Can't I just add the horizontal and vertical distances?" Let's compare:

Counting Squares Distance Formula
Method Add \( \text{horizontal} + \text{vertical} \) \( \sqrt{(\Delta x)^2+(\Delta y)^2} \)
Path L-shaped (two turns) Straight line
Result for \( (0,0) \) to \( (3,4) \) 7 units 5 units
Gives true distance? No: overestimates Yes

How to Apply the Formula

1
Identify coordinates: label the two points as \( (x_1, y_1) \) and \( (x_2, y_2) \).
2
Subtract x-values: compute \( x_2 - x_1 \) for the horizontal change.
3
Subtract y-values: compute \( y_2 - y_1 \) for the vertical change.
4
Square both differences, then add them together.
5
Take the square root of the sum. Simplify if possible.

Example 1: (1, 2) to (4, 6)

\(x\)
\(y\)
1
2
3
4
5
6
1
2
3
4
5
6
(1, 2)
(4, 6)
3
4
\(d = ?\)

Example 2: (-2, 3) to (4, -1)

\(x\)
\(y\)
1
2
3
4
-1
-2
-3
3
4
-1
-2
(-2, 3)
(4, -1)
6
4
\(d = ?\)

Example 3: (0, 0) to (5, 12)

This is a classic Pythagorean triple: 5, 12, 13. Let's verify with the formula:

1
Identify: \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (5, 12) \).
2
Subtract: \( x_2 - x_1 = 5 \), \( y_2 - y_1 = 12 \).
3
Square and add: \( 5^2 + 12^2 = 25 + 144 = 169 \).
4
Square root: \( \sqrt{169} = \) \(13\).

Common Pythagorean Triples

A Pythagorean triple is a set of three positive integers \( (a, b, c) \) that satisfy \( a^2 + b^2 = c^2 \). Memorizing a few will speed up your work.

Leg \( a \) Leg \( b \) Hypotenuse \( c \)
3 4 5
5 12 13
8 15 17
7 24 25
6 8 10

When the Answer is Not a Whole Number

Most distances between points are not perfect integers. You can leave your answer as a simplified radical or round to a decimal.

1
\( d = \sqrt{(7-2)^2 + (4-1)^2} \)
2
\( d = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \)
3
\( \sqrt{34} \) cannot be simplified further. As a decimal: \( \approx 5.83 \)

Real-World Applications

The distance formula appears in many fields beyond math class:

Navigation
GPS systems calculate straight-line distances between waypoints using coordinates.
Video Games
Games use the distance formula to detect collisions and measure how far apart characters are.
Construction
Builders calculate diagonal measurements across rectangular spaces to ensure walls are square.

Common Mistakes to Avoid

✕ Forgetting to Square
Students write \( \sqrt{(x_2 - x_1) + (y_2 - y_1)} \) instead of squaring each difference first. Always square before adding.
✕ Sign Errors from Subtraction
Forgetting that squaring cancels out negative signs. For example, \( (2-5)^2 = (-3)^2 = 9 \). If you forget to square, you might mix up signs or try to take the square root of a negative number.
✕ Forgetting the Square Root
Computing \( (x_2-x_1)^2 + (y_2-y_1)^2 \) gives \( d^2 \), not \( d \). You must take the square root at the end.

Summary: Key Takeaways

Concept Key Fact
Pythagorean Theorem \( a^2 + b^2 = c^2 \)
Distance Formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Horizontal Leg \( |x_2 - x_1| \)
Vertical Leg \( |y_2 - y_1| \)
Common Triples \( (3,4,5), (5,12,13), (8,15,17) \)
When not a whole number Simplify the radical or approximate

Question 1 of 5

What is the distance between the points \( (0, 0) \) and \( (3, 4) \)?

Question 2 of 5

Find the distance between \( (1, -2) \) and \( (4, 2) \).

Question 3 of 5

What is the distance between \( (-3, 1) \) and \( (2, 4) \)?

Question 4 of 5

A student computed the distance between \( (2, 5) \) and \( (6, 8) \) as \( \sqrt{(6-2) + (8-5)} = \sqrt{7} \). What mistake did they make?

Question 5 of 5

Two points are 5 units apart horizontally and 12 units apart vertically. What is the distance between them?