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Pythagorean Theorem, Distance Formula, and Equation of a Circle

July 19, 2010 GB High School Geometry, High School Mathematics

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. » Read more

11 comments conic sections equation of a circle, derive equation of a circle, distance formula, equation of a circle, equation of a parabola, find equation of a circle, hypotenuse of a right triangle, pythagorean theorem, pythagorean theorem example, pythagorean theorem formula, pythagorean theorem proof, right triangles

Rational and Irrational Numbers

May 5, 2010 GB College Mathematics, High School Algebra, High School Mathematics, Number Theory

The need of men to perform certain mathematical operations led to the birth of different types of numbers. People in the ancient times used only counting numbers to keep track of the number of their belongings such as animals. The concept of trade led to the invention of 0 and negative numbers. The need to divide led to the invention of rational numbers.

In this article, we are going to take a look at the characteristics of rational and irrational numbers.

Rational numbers are numbers of the form a/b where a, b are integers, and b not equal to zero. Rational numbers are called rational not because they are reasonable, but because they are a ratio of two integers. It is worthy to note the conditions in the definition. That is because not all fractions are rational numbers. For example, $\frac{2 \pi}{3 \pi}$ is a fraction, but it is only a rational number when simplified. From the definition, we can deduce that all integers are rational numbers since {…,-3, -2, -1, 0, 1, 2, 3,…} = {…, -3/1, -2/1, -1/1, 0/1, 2/1, 3/1, …}.

Geometric Interpretation of Rational and Irrational Numbers

In ancient times the Greeks, particularly the Pythagoreans, believed that all quantities are rational; that is, all quantities can be expressed as a ratio of two integers. Geometrically, this can be interpreted as follows. Given any two lengths, a unit length can be found that can measure the two lengths exactly without gaps or overlaps. In the example in Figure 1, we have two segments a and b, and we found a unit length that would fit exactly a whole number of times in both segments. The ratio of a:b is 7:6, or we can express it in a fraction that a is 7/6th of b.

Figure 1 – The division of segments a and b into unit lengths.

This belief, as most of us now know, was proven to be false. The Pythagoreans later discovered that given a square with length 1 unit, no unit length, however short, can be found to measure both the side of the square and its diagonal like what we have done above. They have concluded that the length of the diagonal cannot be expressed as the ratio of two integers and hence not rational. Today, numbers that are not rational are called irrational numbers. Hence, we define irrational numbers as numbers that cannot be expressed as a ratio of two integers.

Figure 2 – The square with length 1 unit has irrational diagonal.

Using the Pythagorean Theorem, we now know that the length of the diagonal of a square with side length 1 unit is equal to $\sqrt{2}$ . We have already discussed and proved that $\sqrt{2}$ is irrational.

The collection of all rational and irrational numbers is called real numbers. Geometrically, real numbers are represented by the real line as shown in Figure 3.

Figure 3 – The real number line.

Each real number can be represented by a point on the real number line and every point on the number line has a corresponding real number.

Another Representation of Rational and Irrational Numbers

Aside from fractions, we can also represent rational numbers with decimals. For example, 1/5 = 0.2 and 1/3 = 0.333…. Observe that 0.2 has a finite number of decimals while 0.333… has infinite. Irrational numbers can also be represented using decimals. They are the types of decimals that do not end and do not repeat.

Several irrational numbers are very popular, and we had been using them from elementary school to college. The irrational numbers $\pi$ , $e$ and $\phi$ are several of irrational numbers that we are acquainted with.

Figure 4 – The structure of the real number system.

From our discussion above, we can see that real numbers are divided into two main subsets – rational and irrational numbers.

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The Algebraic and Geometric Proofs of Pythagorean Theorem

February 3, 2010 GB High School Geometry, High School Mathematics

The Pythagorean Theorem states that if a right triangle has side lengths $a, b$ and $c$ , where $c$ is the hypotenuse, then the sum of the squares of the two shorter lengths is equal to the square of the length of the hypotenuse.

Figure 1 – A right triangle with side lengths a, b and c.

Putting it in equation form, we have

$a^2 + b^2 = c^2$ .

For example, if a right triangle has side lengths $5$ and $12$ , then the length of its hypotenuse is $13$ , since $c^2 = 5^2 + 12^2 \Rightarrow c = 13$ .

Exercise 1: What is the hypotenuse of the triangle with sides $1$ and $\sqrt{3}$ ?

The converse of the theorem is also true. If the side lengths of the triangle satisfy the equation $a^2 + b^2 = c^2$ , then the triangle is right. For instance, a triangle with side lengths $(3, 4, 5)$ satisfies the equation $3^2 + 4^2 = 5^2$ , therefore, it is a right triangle.

Geometrically, the Pythagorean theorem states that in a right triangle with sides $a, b$ and $c$ where $c$ is the hypotenuse, if three squares are constructed whose one of the sides are the sides of the triangle as shown in Figure 2, then the area of the two smaller squares when added equals the area of the largest square.

Figure 2 – The geometric interpretation of the Pythagorean theorem states that the area of the green square plus the area of the red square is equal to the area of the blue square.

One specific case is shown in Figure 3: the areas of the two smaller squares are $9$ and $16$ square units, and the area of the largest square is $25$ square units.

Exercise 2: Verify that the area of the largest square in Figure 3 is 25 square units by using the unit squares.

Figure 3 – A right triangle with side lengths 3, 4 and 5.

Similarly, triangles with side lengths $(7, 24, 25)$ and $(8, 15, 17)$ are right triangles. If the side lengths of a right triangle are all integers, we call them Pythagorean triples. Hence, $(7, 24, 25)$ and $(8, 15, 17)$ are Pythagorean triples.

Exercise 3: Give other examples of Pythagorean triples.

Exercise 4: Prove that there are infinitely many Pythagorean triples.

Proofs of the Pythagorean Theorem

There are more than 300 proofs of the Pythagorean theorem. More than 70 proofs are shown in tje Cut-The-Knot website. Shown below are two of the proofs. Note that in proving the Pythagorean theorem, we want to show that for any right triangle with hypotenuse $c$ , and sides $a$ , and $b$ , the following relationship holds: $a^2 + b^2 = c^2$ .

Geometric Proof

First, we draw a triangle with side lengths $a, b$ and $c$ as shown in Figure 1. Next, we create 4 triangles identical to it and using the triangles form a square with side lengths $a + b$ as shown in Figure 4-A. Notice that the area of the white square in Figure 4-A is $c^2$ .

Figure 4 – The Geometric proof of the Pythagorean theorem.

Rearranging the triangles, we can also form another square with the same side length as shown in Figure 4-B.This means that the area of the white square in the Figure 4-A is equal to the sum of the areas of the white squares in Figure 4-B (Why?). That is, $c^2 = a^2 + b^2$ which is exactly what we want to show. *And since we can always form a (big) square using four right triangles with any dimension (in higher mathematics, we say that we can choose arbitrary $a$ and $b$ as side lengths of a right triangle), this implies that the equation $a^2 + b^2 = c^2$ stated above is always true regardless of the size of the triangle.

Exercise 5: Prove that the quadrilateral with side length C in Figure 4-A is a square.

Algebraic Proof

In the second proof, we will now look at the yellow triangles instead of the squares. Consider Figure 4-A. We can compute the area of a square with side lengths $a + b$ using two methods: (1) we can square the side lengths and (2) we can add the area of the 4 congruent triangles and then add them to the area of the white square which is $c^2$ . If we let $A$ be the area of the square with side $b + a$ , then calculating we have

Method 1: $A = (b + a)^2 = b^2 + 2ab +a^2$

Method 2: $A = 4(1/2ab) + c^2 = 2ab + c^2$

Methods 1 and 2 calculated the area of the same square, therefore they must be equal. This means that we can equate both expressions. Equating we have,

$b^2 + 2ab + a^2 = 2ab + c^2 \Rightarrow a^2 + b^2 = c^2$

which is exactly what we want to show.

Leave a comment algebraic proofs, geometric proofs, proof of pythagorean theorem, pure mathematics, pythagora's theorem, pythagorean theorem, pythagorean triples, right heron triangles, right triangles

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