**Unformatted text preview: **Q are arranged vertically or horizontally, or describe the exact same point, we
cannot use the above geometric argument to derive the distance formula. It is left to the reader to
verify Equation 1.1 for these cases.
Example 1.1.3. Find and simplify the distance between P (−2, 3) and Q(1, −3).
Solution.
(x2 − x1 )2 + (y2 − y1 )2 d= (1 − (−2))2 + (−3 − 3)2
√
9 + 36
=
√
=35
= √
So, the distance is 3 5.
Example 1.1.4. Find all of the points with x-coordinate 1 which are 4 units from the point (3, 2).
Solution. We shall soon see that the points we wish to ﬁnd are on the line x = 1, but for now
we’ll just view them as points of the form (1, y ). Visually,
y
3 (3, 2) 2
1 distance is 4 units
2 3 x −1 (1, y )
−2
−3 We require that the distance from (3, 2) to (1, y ) be 4. The Distance Formula, Equation 1.1, yields 8 Relations and Functions d= (x2 − x1 )2 + (y2 − y1 )2 4=
4= (1 − 3)2 + (y − 2)2
4 + (y − 2)2 42 =
16
12
(y − 2)2
y−2
y−2
y =
=
=
=
=
= 4 + (y − 2)2 2 4 + (...

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