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DISTANCE BETWEEN TWO POINTS The
distance between two points, P, and P_{2}, 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 P_{2} be (X_{2},Y_{2}), as shown in
figure 12. By the Pythagorean theorem,
Figure 12.Distance between two points. where
P_{1}N represents the distance
between x, and x2, P_{2}N represents the distance between y1 and y2, and d represents
the distance from P_{1} to P_{2}. We can express the length of
P_{1}N in terms of x, and x_{2} and the length of P_{2}Nin
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 12, x, = 2,x_{2} =
6, y, = 2, and y2 = 5. Find the
length of d. SOLUTION:
This
result could have been foreseen by observing that triangle P_{1}NP_{2}
is a 345 triangle. EXAMPLE: Find the distance between P_{1
}(4,6) and P_{2} (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 P_{1} and P_{2}. In
figure 13, P is a point lying on the line joining P_{1} and P_{2}
so that
If
P should lie 1/4 of the way between P_{1} and P_{2}, then k would equal 1/4. Triangles
P_{1}MP and P_{1}NP_{2} are similar. Therefore,
Figure 13.Division of a line segment.
Since
is the ratio that defines k_{1} then ^{
} Therefore, P_{1}M =
k(PIN) Refer
again to figure 13 and observe that P_{1}N is equal to X2^{} x,. Likewise, P_{1}M is equal
to x  x_{1}. When you replace P_{1}M and P_{1}N 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 P_{1} and P_{2} are
determined by the value of k. EXAMPLE: Find the coordinates of a point 1/4 of the way from P_{1}(2,3)
to P_{2}(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 P_{1}(2,4)
and P_{2}(4,6). SOLUTION:
Therefore, the midpoint is (3,5). 