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DISTANCE BETWEEN TWO POINTS The
distance between two points, P, and P2, can be expresssed in terms
of their coordinates by using the Pythagorean theorem. From your study of
Mathematics, Volume 1, you should recall that this theorem is stated as
follows: In
a right triangle, the square of the length of the hypotenuse (longest side) is
equal to the sum of the squares of the lengths of the other two sides.
Let
the coordinates of P, be (x),y,) and let those of P2 be (X2,Y2), as shown in
figure 1-2. By the Pythagorean theorem,
Figure 1-2.-Distance between two points. where
P1N represents the distance
between x, and x2, P2N represents the distance between y1 and y2, and d represents
the distance from P1 to P2. We can express the length of
P1N in terms of x, and x2 and the length of P2Nin
terms of y1 and y2 as follows:
Although
we have demonstrated the formula for the first quadrant only, it can be proven
for all quadrants and all pairs of points. EXAMPLE: In figure 1-2, x, = 2,x2 =
6, y, = 2, and y2 = 5. Find the
length of d. SOLUTION:
This
result could have been foreseen by observing that triangle P1NP2
is a 3-4-5 triangle. EXAMPLE: Find the distance between P1
(4,6) and P2 (10,4). SOLUTION:
DIVISION
OF A LINE SEGMENT Many
times you may need to find the coordinates of a point that is some known
fraction of the distance between P1 and P2. In
figure 1-3, P is a point lying on the line joining P1 and P2
so that
If
P should lie 1/4 of the way between P1 and P2, then k would equal 1/4. Triangles
P1MP and P1NP2 are similar. Therefore,
Figure 1-3.-Division of a line segment.
Since
Therefore, P1M =
k(PIN) Refer
again to figure 1-3 and observe that P1N is equal to X2- x,. Likewise, P1M is equal
to x - x1. When you replace P1M and P1N with their equivalents in terms of x, the preceding equation
becomes
By
similar reasoning,
The
x and y found as a result of the foregoing discussion are the coordinates of
the desired point, whose distances from P1 and P2 are
determined by the value of k. EXAMPLE: Find the coordinates of a point 1/4 of the way from P1(2,3)
to P2(4,1). SOLUTION:
Therefore, point P is
When the midpoint of a line segment is to be
found, the value of (Io is 1/2. Therefore,
such that
By similar reasoning,
EXAMPLE: Find the midpoint of the line between P1(2,4)
and P2(4,6). SOLUTION:
Therefore, the midpoint is (3,5). |
 
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is the ratio that defines 
.
