The Straight Line: Use Pythagoras’ Theorem to Calculate Distance Between Two Points

Understanding the Concept

• Pythagoras’ Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be represented as `a² = b² + c²`.

• When dealing with coordinates, treat each point as one corner of a right-angled triangle, and the distance between them as the hypotenuse.

• The distance d between two points (x₁, y₁) and (x₂, y₂) can therefore be calculated with this formula remembering it comes directly from the theorem itself: `d = √[(x₂ - x₁)² + (y₂ - y₁)²]`.

Applying the Formula

• First, determine the coordinates of the two points you are calculating distance between.

• Subtract the x-coordinates and subtract the y-coordinates of these two points. This gives you the lengths of the other two sides of your right-angled triangle.

• Square these lengths (raise to the power of 2).

• To apply this to a problem, consider the points A(2,3) and B(6,7). The distance between A and B is: `d = √[(6 - 2)² + (7 - 3)²] = √[16 + 16] = √32`.
• Remember, the numbers representing the coordinates can be negative, in which case you may need to subtract a negative (which effectively becomes addition). For example, for the points A(-2, -3) and B(1, 1), the distance is: `d = √[(1 - -2)² + (1 - -3)²] = √[9 + 16] = √25 = 5`.