prove the distance formula
using Pythagoras theorem
Answers
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If 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.