Transcript Coordinates, points and lines

Coordinates, points and lines
Distance between two points
Using the cartesian coordinate system, the distance
between two points A and B is the length of the line
segment AB.
Using Pythagoras Theorem
The distance between the points ( x1 , y1 ) and ( x2 , y2 )
(or the length of the line segment joining them) is
 x2  x1    y2  y1 
2
2
.
The mid-point of a line segment
The mid-point of a line segment joining  x1 , y1  and ( x2 , y2 )
has coordinates
1
1

x

x
,
y

y




1
2
1
2 .

2
2


Do Exercise 1A, Q.14, Q.16, Q.20, Q,22, p.7
The gradient of a line segment
Unlike distance and the mid-point, the gradient is a property
of the whole line, not just of a particular line segment.
If you take any two points and find the increase in the x 
and y  coordinates as you go from one to the other, then
the fraction
y  step
x  step
is called the gradient of the line.
The gradient of the line joining  x1 , y1  to  x2 , y2  is
y2  y1
.
x2  x1
Read Example 1.3.1, 1.3.2, pp.5-6
The equation of a line
We can tell whether or not a point is on a line (or curve) by
substituting the coordinates into the equation and seeing if the
equation is satisfied by the coordinates of the point.
The equation of a line or curve is a rule for determining
whether or not the point with coordinates  x, y  lies on
the line or curve.
To find the equation of the line with gradient m through
the point A with coordinates  x1 , y1  consider the general
point P with coordinates ( x, y ). The gradient of AP is
y  y1
.
x  x1
The equation of the line through  x1 , y1  with gradient m is
y  y1  m  x  x1  .
Recognising the equation of a
straight line
It is easy to see that the line equation y  y1  m  x  x1  can
be rewritten in the form
y  mx  c,
and that the point  0,c  lies on the y-axis.
The number c is called the y - intercept of the line.
To find the x-intercept, put y  0 in the equation, which gives
c
x   , where m  0.
m
If m  0, the line has equation y  c.
Recognising the equation of a
straight line
Straight lines parallel to the y -axis have equations of the
form x  k.
Note that the line x  k does not have a gradient;
its gradient is undefined. Its equation cannot be
written in the form y  mx  c.
To find the intersection of two (non-parallel) straight lines
simply solve the associated simultaneous equations.
Do Exercise 1B, Q.5-Q.13, p.12
The gradients of perpendicular
lines
If the gradient of a line segment PB is m, you
can draw a 'gradient triangle' PAB in which PA
is one unit and AB is m units.
If the gradient triangle PAB is rotated through a
right-angle to PAB, then
gradient of PB 
Two lines with gradients m1 and m2 are perpendicular if
m1m2  1,
or
m1  
1
,
m2
or
m2  
Do Exercise 1C, pp. 14-15
Do Misc. Exercise 1, Q.13-Q.22, p.16
1
.
m1
y -step
1
 .
x-step
m