The pythagorean theorem. Topics in trigonometry.

THE PYTHAGOREAN THEOREM

BASIC TO TRIGONOMETRY is the theorem that relates the sides of a right-angled triangle.(A theorem is a statement that can be proved). Credit for this theorem goes to the Greek philosopher Pythagoras, who lived in the 6th century B. C.

Geometric representation

In a right triangle the square drawn on the side opposite
the right angle is equal to the squares drawn on the sides that make
the right angle.

This means that if ABC is a right triangle with the right angle at A, then the square drawn on BC opposite the right angle, is equal to the two squares together on CA, AB.

In other words, if it takes one can of paint to paint the square on BC, then it will also take exactly one can to paint the other two squares.

This is how the sides of a right triangle are related -- by the squares drawn on them -- and we can easily illustrate it with numbers.

The square drawn on the side opposite the right angle, 25, is equal to the squares on the sides that make the right angle: 9 + 16.

The side opposite the right angle is called the hypotenuse ("hy-POT'n-yoos"; which literally means stretching under).

We will indicate the right angle by a small square.

As for the sides themselves, they will be the square roots. That is, when the squares are 9, 16, 25, then the sides will be 3, 4, 5. (The square root of 9 is 3; the square root of 16 is 4; and the square root of 25 is 5. See Topic 2: Radicals.)

Problem 1. How big is the remaining square? And how long are the sides of the triangle?

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Problem 2. How big is the remaining square? And how long are the sides of the triangle?

Algebraic representation

Algebraically, if the hypotenuse is c, and the sides are a, b, then

c² = a² + b².

Example 1. How long is the unknown side x?

Answer. Here, the hypotenuse is unknown. We have

x² = 8² + 6²

x² = 64 + 36

x² = 100.

Therefore,

x = = 10.

Example 2. How long is the unknown side x?

Answer. Here, the unknown is not the hypotenuse but a side.

x² + 1² = 2²

x² = 4 − 1 = 3./P>

Therefore,

x = .

Problem. In each figure, is the unknown length x the hypotenuse or a side? In each case, find the unknown length x.

a) x squared = 2 squared + 3 squared b) x squared + 5 squared = 13 squared

In figure a), the hypotenuse is unknown. We have
      x² = 2² + 3²
          = 4 + 9
          = 13
       x =

In figure b), it is the side
that is unknown:
x² + 5² = 13²
x² = 169

25
= 144
x =

= 12

A proof of the Pythagorean theorem

The area of a square found in two different ways.

Let a right triangle have sides a, b, and hypotenuse c. And let us arrange four of those triangles to form a square whose side is a + b. (Fig. 1)

Now, the area of that square is equal to the sum of the four triangles, plus the interior square whose side is c.

Two of those triangles taken together, however, are equal to a rectangle whose sides are a, b. The area of such a rectangle is a times b: ab. Therefore the four triangles together are equal to two such rectangles. Their area is 2ab.

As for the square whose side is c, its area is simply c². Therefore, the area of the entire square is

c² + 2ab. . . . . . . (1)

At the same time, an equal square with side a + b (Fig. 2) is made up of a square whose side is a, a square whose side is b, and two rectangles whose sides are a, b. Therefore the area of that square is

a² + b² + 2ab.

But this is equal to the square formed by the triangles, line (1):

a² + b² + 2ab = c² + 2ab.

Therefore, on subtracting the two rectangles -- 2ab -- from each square, we are left with

a² + b² = c².

This is the Pythagorean theorem!

Next Topic: The Definitions of the Trigonometric Functions (Ratios)

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