32 ## THE PYTHAGOREAN DISTANCE FORMULAThe distance of a point from the origin The distance between any two points A proof of the Pythagorean theorem BASIC TO TRIGONOMETRY and calculus is the theorem that relates the squares drawn on the sides of a right-angled triangle. Credit for the proving the theorem goes to the Greek philosopher Pythagoras, who lived in the 6th century B. C. Here is the statement of the theorem: In a right triangle the square drawn on the side opposite the right angle That 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. The side opposite the right angle is called the hypotenuse ("hy-POT'n-yoos"; which literally means stretching under). Algebraically, if the hypotenuse is
For a proof, see below. Problem 1. State the Pythagorean theorem in words. In a right triangle the square on the side opposite the right angle will equal the squares on the sides that make the right angle.
Problem 2. Calculate the length of the hypotenuse To see the answer, pass your mouse over the colored area. a)
b)
Since 9 is a square number, and a common factor of 9 and 36, then we may anticipate simplifying the radical by writing 9 + 36 = 9(1 + 4) = 9 We could, of course, have written 9 + 36 = 45 = 9 The distance According to the Pythagorean theorem, and the meaning of the rectangular coördinates (
Therefore, "The distance of a point from the origin Example 1. How far from the origin is the point (4, −5)? Problem 3. How far from the origin is the point (−5, −12)?
The distance between any two points How far is it from (4, 3) to (15, 8)? Consider the distance Therefore, the horizontal leg of that triangle is simply the distance from 4 to 15: 15 − 4 = 11. The vertical leg is the distance from 3 to 8: 8 − 3 = 5. Therefore, To find a formula, let us use subscripts and label the two points as ( The subscript 1 labels the coördinates of the first point; the subscript 2 labels the coördinates of the second. We write the absolute value because Here then is the Pythagorean distance formula between any two points: It is conventional to denote the difference of
Δ Similarly, Δ Therefore, Example 2. Calculate the distance between the points (1, 3) and (4, 8).
Therefore,
But (−3)² = 9, and (−5)² = 25. The distance between the two points is the same. Example 3. Calculate the distance between the points (−8, −4) and (1, 2).
Therefore, Problem 4. Calculate the distance between (2, 5) and (8, 1)
Problem 5. Calculate the distance between (−11, −6) and (−16, −1)
A proof of the Pythagorean theorem Let a right triangle have sides Now, the area of that square is equal to the sum of the four triangles, plus the interior square whose side is Two of those triangles taken together, however, are equal to a rectangle whose sides are As for the square whose side is
At the same time, an equal square with side
But this is equal to the square formed by the triangles, line (1):
Therefore, on subtracting the two rectangles 2
That is the Pythagorean theorem. Next Lesson: The equation of a straight line Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |