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Geometry:Deriving the Distance Formula

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Deriving the Distance Formula

Distance Formula in Two Dimensions

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Figure 1. Graph of P1 and P2

To solve for distance between two points P1 and P2 , one can see that if we plot two points in two dimensions we can form something like in figure1. From figure 1, we can see that the two points forms a right triangle a point Px whose coordinates are x2,y1. A right triangle is a “triangle with one angle equal to 90 degrees” (Weisstein, 2007). From this right triangle, we can solve the distance (d) if we know the lengths of the other two sides of the triangle P1 to Px (a) and Px to P2 (b) using the Pythagorean theorem.

The Pythagorean theorem “states that   “ (Mahler & Wesner, 1994).

According to Figure 1, we can express a in terms of x2 and x1 while b in terms of y2 and y1:

Substituting eqn1 and eqn2:

Solving for c:

Eqn4 is the distance formula of two points in two dimensions.

Distance Formula in Three Dimensions

In three dimensions where P1 is (x1,y1,z1) and P2 is (x2,y2,z2), the solution comes in two step. First we get the distance using P1(x1,y1) and P2(x2,y2) with our line of site in z-axis then we get c using the distance formula of two dimension

Then the distance c forms another right triangle with the third dimension distance z2-z1 if we rotate the axis and make the line where c lies as one of the axis and apply Pythagorean theorem again.

Substituting eqn 4 to eqn 5 becomes:

Solving for d:

Eqn 7 is the distance formula in three dimensions.

Real Life Applications

            One application of the distance formula is in finding the length of a rope that you need to tie the tip of two post T1 and post T2 of different lengths h1 and h2 together knowing only the length of the posts h1 and h2, and distance between the post X1. Measuring the distance of the tips directly would be difficult and inconvenient because of the need of climbing at the two posts. So that we can use the distance formula by plotting the two post in a rectangular coordinate system as shown in figure2 so that the tip of T1 is at (h1,0) and the tip of T2 is at (h2,X1). Using the distance formula, the length L can be calculated as

Type equation here.

Figure 2. PostT1 and T2


Weisstein, Eric W. “Right Triangle” . Retrieved from the MathWorld-A Wolfram Web Resource: http://mathworld.wolfram.com/RightTriangle.html,  on March 19, 2007

Wesner, T. H. & Mahler, P.H. (1994). ”Some Geometry”. In College Algebra & Trigonometry with Applications. (pp. 310-311). IOWA : Wm. C. Brown Communications, Inc.

Cite this Geometry:Deriving the Distance Formula

Geometry:Deriving the Distance Formula. (2016, Jun 23). Retrieved from https://graduateway.com/geometryderiving-the-distance-formula/

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