The distance formula and the midpoint formula
The distance formula
.
Given two points
A
(
)
,
x
1
y
1
and
B
(
)
,
x
2
y
2
in the coordinate plane where, in the first instance,
B
is above and to the right of
A
as in the
picture, the horizontal distance between the two points is

x
2
x
1
and the vertical distance between the two points is

y
2
y
1
.
If the distance between the two points
A
and
B
is
d
, applying Pythagoras' theorem in the rightangled triangle
ABC
, where
AC
is parallel to
the
x
axis and
BC
is parallel to the
y
axis, it follows that
=
d
2
+
(
)

x
2
x
1
2
(
)

y
2
y
1
2
.
Similar diagrams can be drawn for the other possible configurations for
A
and
B
.
If
B
is to the right and below
A
, then the horizontal
distance between
A
and
B
is still

x
2
x
1
but the vertical distance is

y
1
y
2
.
Because
=
(
)

y
1
y
2
2
(
)

y
2
y
1
2
, applying Pythagoras' theorem in the triangle
ABC
still gives
=
d
2
+
(
)

x
2
x
1
2
(
)

y
2
y
1
2
.
The other two possible configurations with
B
to the left of
A
also give the same result. Hence in all cases the distance d between
A
and
B
is
given by
=
d
+
(
)

x
2
x
1
2
(
)

y
2
y
1
2
.
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 Spring '08
 Uri
 Distance Formula, Pythagorean Theorem, triangle

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