Answer:
The Pythagorean theorem states that with a right-angled triangle, the sum of the squares of the two sides that form the right angle is equal to the square of the third, longer side, which is called the hypotenuse.
f we consider what the distance formula really tells you, we can see the similarities. It is more than just a similar form.
The distance formula is commonly seen as:
D
=
√
(
x
1
−
x
2
)
2
+
(
y
1
−
y
2
)
2
We commonly write the Pythagorean Theorem as:
c
=
√
a
2
+
b
2
Consider the following major points (in Euclidean geometry on a Cartesian coordinate axis):
The definition of a distance from
x
to
±
c
is
|
x
−
c
|
.
There is the relationship where
√
(
x
−
c
)
2
=
|
x
−
c
|
=
x
−
c
AND
−
x
+
c
The distance from one point to another is the definition of a line segment.
Any diagonal line segment has an
x
component and a
y
component, due to the fact that a slope is
Δ
y
/
Δ
x
. The greater the
y
contribution, the steeper the slope. The greater the
x
contribution, the flatter the slope.
What do you see in these formulas? Have you ever tried drawing a triangle on a Cartesian coordinate system? If so, you should see that these are two formulas relating the diagonal distance on a right triangle that is composed of two component distances
x
and
y
.
Or, we could put it another way through substitutions based on the distance definitions above. Let:
x
1
−
x
2
=
±
a
y
1
−
y
2
=
±
b
(depending on if
x
1
>
x
2
or
x
1
<
x
2
, and similarly for
y
.)
Now what do you see? An equivalence.
D
=
√
(
±
a
)
2
+
(
±
b
)
2
=
c
=
√
a
2
+
b
2
In short, the distance formula is a formalization of the Pythagorean Theorem using
x
and
y
coordinates. In other words, they are the same thing in two seemingly different contexts.
