How does the distance formula ensure that the distance between two different points is positive?

How does the distance formula ensure that the distance between two different points is positive? – The distance formula is a fundamental mathematical tool used to calculate the straight-line distance between two points in a coordinate system. The formula ensures that the distance between any two points is always positive or zero (in the case where the two points are the same). Let’s explore how this is guaranteed through a breakdown in a table, followed by detailed explanations and a conclusion.

How does the distance formula ensure that the distance between two different points is positive?

Component	Symbol	Description
Coordinates of Point 1	(x1,y1)(x_1, y_1)(x1​,y1​)
Coordinates of Point 2	(x2,y2)(x_2, y_2)(x2​,y2​)
Difference in X Coordinates	Δx\Delta xΔx	The difference between the x coordinates: Δx=x2−x1\Delta x = x_2 – x_1Δx=x2​−x1​.
Difference in Y Coordinates	Δy\Delta yΔy	The difference between the y coordinates: Δy=y2−y1\Delta y = y_2 – y_1Δy=y2​−y1​.
Distance Formula	ddd	d=(Δx)2+(Δy)2d = \sqrt{(\Delta x)^2 + (\Delta y)^2}d=(Δx)2+(Δy)2​

Detailed Explanations:

1. Components of the Distance:
   • Difference in X Coordinates (Δx\Delta xΔx) and Difference in Y Coordinates (Δy\Delta yΔy): These components represent the horizontal and vertical distances between the two points. The differences are squared in the distance formula.
2. Squaring the Differences:
   • Squaring Δx\Delta xΔx and Δy\Delta yΔy ensures that these values are non-negative. Squaring any real number, whether positive or negative, results in a non-negative value.
3. Summing the Squares:
   • Adding two non-negative numbers ((Δx)2(\Delta x)^2(Δx)2 and (Δy)2(\Delta y)^2(Δy)2) will always yield a non-negative result. This summation forms the basis of the distance calculation under the radical.
4. Square Root Function:
   • The square root function, represented by ⋅\sqrt{\cdot}⋅​, only outputs non-negative results for non-negative inputs. Since the input here is the sum of two squared terms, the output (distance) must be non-negative.

Conclusion:

The distance formula, d=(Δx)2+(Δy)2d = \sqrt{(\Delta x)^2 + (\Delta y)^2}d=(Δx)2+(Δy)2​, inherently guarantees that the distance calculated between any two points in a coordinate system is non-negative (positive or zero). This assurance stems from the properties of the arithmetic operations used:

• Squaring the coordinate differences eliminates any negative signs, ensuring that the values used in further calculations are non-negative.
• Summing these squares maintains non-negativity, preparing a suitable non-negative value for the final operation.
• Taking the square root of a non-negative value (the sum of the squares) ensures the result remains non-negative, reflecting the real-world physical concept of distance, which cannot be negative.

Thus, the distance formula is robust in ensuring that the calculated distances are always meaningful within the context of spatial measurements.