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THE DISTANCE FORMULA
Jump right to the exercises!
The distance between points
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
is given by the formula
(
x
2
-
x
1
)
2
+
(
y
2
-
y
1
)
2
.
Here,
x
1
(read as "x sub 1") denotes the x-coordinate of the first point,
and
y
1
(read as "y sub 1")
denotes the y-coordinate of the first point.
Similarly,
x
2
denotes the x-value of the second point,
and
y
2
denotes the y-value of the second point.
Notice that
(a-b)
2
is equal to
(b-a)
2
.
In words, to find the distance between two points, do the following:
- Subtract the x-values of the two points (in any order), and square this result.
- Subtract the y-values of the two points (in any order), and square this result.
- Add together the previous two quantities.
- Take the square root of the result.
The Distance Formula follows easily from the Pythagorean Theorem, as suggested by the
picture below:
Use the Geometer's Sketchpad
(hands-on geometry at its best!) to explore the
distance formula.
EXAMPLES:
Question: Find the distance between (1,-3) and (-2,5).
Solution:
(
-
2
-
1
)
2
+
(
5
-
(
-
3
)
)
2
=
73
Question: Does the following formula represent the distance between points (a,b) and (c,d) ?
Answer YES or NO.
(
a
-
c
)
2
+
(
d
-
b
)
2
Solution: YES.
The order that you subtract the numbers does not affect the result.
Question: Does the following formula represent the distance between points (a,b) and (c,d) ?
Answer YES or NO.
(
a
-
b
)
2
+
(
c
-
d
)
2
Solution: NO.
You must subtract the x-values (y-values) of the two points.
NOTE: In the solutions, radicals are reported both unsimplified, and also in simplest form.
For example,
8
is also re-written as
2
2
.
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.