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# 4.3 The Pythagorean Theorem and Distance.

Even though you are probably familiar with the Pythagorean theorem, the result is so important and non-obvious that I am including a proof of it.

4.22   Theorem (Pythagorean Theorem.) In any right triangle, the square on the hypotenuse is equal to the sum of the squares on the two legs.

Proof: Consider a right triangle whose legs have length and , and whose hypotenuse has length , and whose angles are and as shown in the figure.

We have since is a right triangle.

Construct a square with sides of length , and find points dividing the sides of into pieces of sizes and as shown in figure 1. Draw the lines , and , thus creating four triangles congruent to (i.e., four right triangles with legs of length and ). Each angle of is so is a square of side . The four triangles in figure 1 each have area , so
 (4.23)

or

and hence
 (4.24)

The proof just given uses a combination of algebra and geometry. I will now give a second proof that is completely geometrical.

Construct a second square with sides of length , and mark off segments and of length as shown in figure 2. Then draw perpendicular to and let intersect at , and draw perpendicular to and let intersect at . Then is a right angle, since the other angles of the quadrilateral are right angles. Similarly angle is a right angle. Thus is a rectangle so and similarly is a rectangle and . Moreover and are perpendicular since and . Thus the region labeled is a square with side and the region labeled is a square with side .

In figure 2 we have , and hence

 (4.25)

We have since and are both squares with side . Hence from equations (4.23) and (4.25) we see that

Although the theorem we just proved is named for Pythagoras (fl. 530-510 B.C) , it was probably known much earlier. There is evidence that it was known to the Babylonians circa 1000 BC[27, pp 118-121]. Legend has it that

Emperor Yu[circa 21st century B.C.] quells floods, he deepens rivers and streams, observes the shape of mountains and valleys, surveys the high and low places, relieves the greatest calamities and saves the people from danger. He leads the floods east into the sea and ensures no flooding or drowning. This is made possible because of the Gougu theorem [47, page 29].

Gougu'' is the shape shown in the figure, and the Gougu theorem is our Pythagorean theorem. The prose style here is similar to that of current day mathematicians trying to get congress to allocate funds for the support of mathematics.

Katyayana(c. 600 BC or 500BC??) stated the general theorem:

The rope [stretched along the length] of the diagonal of a rectangle makes an [area] which the vertical and horizontal sides make together.[27, page 229]

4.26   Theorem (Distance formula.) If and are points in then the distance from to is

Proof: Draw the vertical line through and the horizontal line through . These lines intersect at the point . The length of is and the length of is and is the hypotenuse of a right angle with legs and .

By the Pythagorean theorem,

so length .

4.27   Notation ( , distance ) If and are points in , I will denote the distance from to by either distance or by .

4.28   Definition (Circle.) Let be a point in , and let . The circle with center and radius is defined to be

The equation

is called the equation of the circle . The circle is called the unit circle.

We will now review the method for solving quadratic equations.

4.29   Theorem (Quadratic formula.) Let , , and be real numbers with .

If , then the equation has no solutions in R.

If , then the set of solutions of the equation is

 (4.30)

The set (4.30) contains one or two elements, depending on whether is zero or positive.)

Proof: Let be real numbers with . Let . Then

Hence has no solutions unless . If , then the solutions are given by

i.e.,

4.31   Example. Describe the set .
The sketch suggests that this set will consist of two points in the first quadrant. Let be a point in the intersection. Then
 (4.32)

and
 (4.33)

It follows that , or or
 (4.34)

(The line whose equation is is shown in the figure. We've proved that the intersection is a subset of this line.) Replace by in equation (4.33) to obtain

i.e.,

i.e.,

i.e.,

By the quadratic formula, it follows that

By equation (4.34)

We have shown that if , then
. It is easy to verify that each of the two calculated points satisfies both equations  (4.32) and (4.33) so

Next: 5. Area Up: 4. Analytic Geometry Previous: 4.2 Reflections, Rotations and   Index
Ray Mayer 2007-09-07