Pythagorean Theorem, Distance Formula, and Equation of a Circle

In my Algebraic and Geometric Proof of the Pythagorean Theorem post, we have learned that a right triangle with side lengths $a$ and $b$ and hypotenuse length $c$, the sum of the squares of $a$ and $b$ is equal to the square of $c$. Placing it in equation form we have $c^2 = a^2 + b^2$.

If we place the triangle in the coordinate plane, having $A$ and $B$ coordinates of $(x_1,y_1)$ and $(x_2,y_2)$ respectively, it is clear that the length of $AC$ is $|x_2 - x_1|$ and the length of $BC$ is $|x_2 - x_1|$.  We are finding the length, which means that we want a positive value; the absolute value signs guarantee that the result of the operation is always positive. But in the final equation,$c^2 = |x_2 - x_1|^2 + |y_2-y_1|^2$, the absolute value sign is not needed since we squared all the terms, and squared numbers are always positive. Getting the square root of both sides we have,

$c = \sqrt{|x_2 - x_1|^2 + |y_2-y_1|^2}$

We say that $c$ is the distance between $A$ and $B$, and we call the formula above, the distance formula.

If we want coordinates of $B(x,y)$ where $x$ and $y$ are variables and the distance of $B$ from $A$ constant, say $r$,  then moving point $B$ about point $D$ maintaining the distance $r$ forms a circle. If $A$ has coordinates $(h,k)$, then $r^2 = (x-h)^2 + (y-k)^2$ which means that $r = \sqrt{ x-h)^2 + (y-k)^2}$.

Observe that the two  equations above are all of the same form, they are all consequences of the  Pythagorean Theorem.  The examples are probably very elementary, but it shows one of the rare beauties of mathematics — the strong connections between and among different concepts.

11 thoughts on “Pythagorean Theorem, Distance Formula, and Equation of a Circle”

1. Pingback: Tips From Tim: The keys to adding distance with irons

2. Way cool. Did you have students discover these connections using computer software? I would think that may help them “retain” the information and hopefully get “excited” about the “power” of mathematics as a universal language.
Lonnise

3. Thanks Lonnise. I have tried GeoGebra before. We can show that the rotating point about a fixed point can be constructed using different right triangles. The GeoGebra grid may also help for students to connect the equation of the circle to the distance formula.

4. Good article. But the modulus sign within the square root is really not required, as square of any real number is always positive.

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