Distance Formula
The distance between the points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is given by the formula
d =Ö [(x_{2}x_{1})^{2} + (y_{2}y_{1})^{2}]
Let P(x1, y1) and Q(x2, y2) be the two points. From P, Q draw PL, QM perpendiculars on the
xaxis and PR perpendicular on QM.
Then, PR = LM = x_{2}  x_{1}
RQ = MQ  MR = MQ  LP = y_{2}  y_{1}
From right angled triangle PRQ, PQ^{2} = PR^{2} + RQ^{2}
d^{2} = (x_{2}  x_{1})^{2} + (y_{2}  y_{1})^{2}
d = Ö2  x_{1})^{2} + (y_{2}  y_{1})^{2}]
Distance from the origin
The distance of the point (x_{1}, y_{1}) from the origin is
d = Ö10)^{2} + (y_{1}0)^{2}]
d = Ö1^{2} + y_{1}^{2}]
Example: Find the distance between the points (6, 8), (2, 5).
d^{2} = (x_{2}x_{1})^{2} + (y_{2}y_{1})^{2}
d^{2} = (26)^{2} + [5(8)]^{2}
d^{2} = (4)^{2} + (3)^{2}
d^{2} = 16 + 9 = 25
d = 5
Answer: 5 units
Division or section formula
To find the coordinates of the point which divides internally the line joining two given points in a given ratio.
Let A(x_{1}, y_{1}) and B(x_{2}, y_{2}) be the two given points and P a point on AB which divides it in the given ratio m_{1}: m_{2}. It is required to find the coordinates of P. Suppose they are (x, y). Draw the perpendiculars AL, PM, BN on OX and Ak, PT on PM and BN respectively. Then, from similar triangles APK and PBT we have
AP/PB = AK/PT = KP/TB  1
Now AP: PB = m_{1} : m_{2}, AK = LM = OM  OL = x  x_{1}
PT = MN = MK = MP  LA = y  y1
TB = NB  NT = NB  MP = y2  y
From 1 we have m_{1}/m_{2} = (x  x_{1})/(x_{2}  x) = (y  y_{1})/(y_{2}  y)
The first two relations give m_{1}/m_{2} = (x  x_{1})/(x_{2}  x) or m_{1}x_{2}  m_{1}x = m_{2}x  m_{2}x_{1}
Thus x = (m_{1}x_{2} + m_{2}x_{1})/(m_{1} + m_{2}) , y = (m_{1}y_{2 }+ m_{2}y_{1})/(m_{1} + m_{2})
The coordinates of midpoint are [(x_{1}+x_{2})/2, (y_{1}+y_{2})/2]
Formula for external division
Construct as shown in figure.
From similar triangles APK and PBT,
AP/PB = AK/PT = KP/TB
or m1/m2 = (x  x_{1})/(x  x_{2}) = (y  y_{1})/(y  y_{2})
From the above we get
x = (m_{1}x_{2}  m_{2}x_{1})/(m_{1 } m_{2})
y = (m_{1}y_{2}  m_{2}y_{1})/(m_{1}  m_{2})
Example: Find the coordinates of the point which divides the join of the points (8,9) and (7,4)
a. Internally in the ratio 2:3
b. Externally in the ratio 4:3
Solution:a. The coordinates of the point in the first case
x = (2 x (7) + 3 x 8)/(2+3) = (14 + 24)/5 = 2
y = 2x4+3x9/(2+3) = 35/5 = 7
(2,7)
b. For external division the coordinates are
x = 4x(7)3x8/*43) = 52
y = 4x43x9/(43) = 11
(52,11)
Directions: Solve the following problems.
