### The Distance Between Two Points

```The Distance Between Two
Points
Mr. Dick Gill
Dr. Marcia Tharp
Introduction
When the Wizard of Oz finally gave
the scarecrow his brain the first words
that the scarecrow spoke were: “The
sum of the squares of the legs of an
isosceles right triangle is equal to the
square of the hypotenuse.” He was
quoting a special case of the
Pythagorean Theorem.
The famous Greek mathematician
Pythagoras is credited with the
discovery illustrated on the next slide
concerning the three sides of any right
triangle (a triangle with one angle equal
to 90 degrees). Pythagoras was a
philosopher and a mystic as well as a
mathematician. For more on
Pythagoras go to:
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html
The hypotenuse of a right triangle is
always the longest side, the side opposite the
right angle. The legs are the other two sides.
In the Pythagorean Theorem, it is customary
to use c for the hypotenuse. For an attractive
and simple demo on why this theorem works
go to:
http://www-groups.dcs.st-and.ac.uk/~history/Diagrams/PythagorasTheorem.gif
Consider the following triangle with
legs of 6 inches and 8 inches.
Find the hypotenuse.
See next page.
Solution
a
2
 b
6
2
 8
2
 c
2
 c
2
36  64  c
2
2
100  c
100  c
10  c
2
The hypotenuse then would be 10
inches long. Further work with the
Pythagorean Theorem will require a
review of square roots. As indicated
above, the square root of 100 is 10
because 102 = 100.
2
81  9 since 9  81
2
64  8 since 8  64
2
49  7 since 7  49.
Looking at the roots above, you may conclude
that would be a number between 8 and 7. In fact, is
approximately 7.75. For the rest of this unit you will
need a calculator that does square roots.
For more about square roots go to:
http://forum.swarthmore.edu/~sarah/hamilton/ham.squa
reroots.html
Example 5
Find the missing side in each of
the following triangles.
• Once you have completed the exercise
click to the next slide to check your
Solutions:
a.
6 2  1 22  x 2
36  144  x2
180  x2
x 
b.
1 8 0  1 3.4 2
x 2  72  92
x2  49  81
x2  32
x 
3 2  5 .6 6
c. 1 12  x 2  1 52
1 2 1 x 2  2 2 5
x2  105
x 
1 0 5  1 0.2 5
Now lets see how this works
in a coordinate system.
1. Click on the link below to see how one finds
the distance between two points in a
coordinate system.
2. Experiment by changing the coordinates of
the point P.
3. Write a paragraph to explain how and why
this works to a fellow student.