The Pythagorean Theorem and Coordinate Systems
Introduction
Step 1. Consider two points, such as (1,1) and (4,5), in a coordinate system; right triangles may
not immediately come to mind.
6
(4,5)
5
Y - axis
4
3
2
1
(1,1)
0
0
1
2
3
4
5
X - axis
Step 2. Add coordinate gridlines and , voilá, you have all the right angles and straight lines that
you need to make right triangles. A little connecting the dots accomplishes the task..
6
(4.5)
5
Y - axis
4
3
2
1
(1,1)
0
0
1
2
3
X - axis
4
5
Step 3. By observing and counting we know that the lengths of the legs are 3 and 4.
6
(4.5)
5
Y - axis
4
c=
_______
3
b=4
2
1
(1,1)
a=3
0
0
1
2
3
4
5
X - axis
Step 4. With the Pythagorean Theorem, it is possible to calculate the length of side c.
a2  b2  c 2,so 32  4 2  c 2,so 9 16  c 2,so 25  c 2 .

What number, multiplied by itself (i.e. squared) equals 25? Reflect on this a moment and you
will deduce that the answer is 5.You can do this on your calculator with a single button: the
square root function answers the question, “What number multiplied by itself equals some, given
number?” Hence, 25  c .
Practice

Repeat each of the above 4 steps for the points below. Use quadrille paper for the graphs.
1. (–7, –1) and (5, 4)
2. (–1, 7) and (7, 1)
3. (–2, 3) and (4, –2)
4. Important question #1: how can you calculate the lengths of sides a and b using only the
coordinates of the given points? This is the distance formula!
5. Repeat 1. – 3. using the distance formula.
6. Select any pair of points that you like. Draw the right triangle figure to calculate the distance
between them, and use the distance formula to calculate the distance between them.