|
Geometry
Clement Radcliffe,Contributor
The
solution to the following problems
will complete the review of the aspects
of coordinate geometry, which we shared
during the last three weeks.
PROBLEM
1
The
coordinates of the points L and N
are (5 , 6) and (8 , -2) respectively.
1.
(i) State the coordinates of the midpoint,
M, of the line LN.
(ii)
Calculate the length of the line LN.
(iii)
Calculate the gradient of the line
LN.
(iv)
Determine the equation of the straight
line which is perpendicular to LN
and which passes through the point,
M.
SOLUTION
(i)
Given the points L (5 , 6) and N (8,
-2):
Then M, id-point of LN is =( (x2
+ x1)/2 , (y2
+ y1)/2 ) Substituting:
M
= ( (8 + 5)/2 , (-2 + 6)/2 ) = (
13/2 , 2)
(ii)
The length of LN2 = (x2
- x1)2 + (y2
- y1)2 Substituting:
=
(8 - 5)2 + (-2 - 6)2
=
32 + (-8)2 =
9 + 64 = 73
...
LN = square root of 73
(iii)
M, the gradient of LN = (y2
- y1)/(x2 -
x1) Substituting:
=
(-2 -6)/(8 - 5) = -8/3
(iv)
Let M1 be the gradient
of the line perpendicular to LN.
...
m x m1 = -1
...
m1 x -8/3 = -1
...
m1 = -1 x -3/8 = 3/8
The
Equation of the line with gradient
3/8 and which passes through the point
(13/2 , 2)
is
(y - y1)/(x - x1)
= m.
Substituting
(y
- 2)/(x - 13/2) = 3/8
...
8(y - 2) = 3( (2x - 13)/2 )
...
16(y - 2) = 3(2x - 13)
... 16y - 32 = 6x - 39
...16y - 6x = 32 - 39 = -7
... 16y - 6x = -7
Problem
2
Determine
three
different features of the equation
2y + 3x = 5
SOLUTION
Given
the
Equation 2y + 3x = 5
...
2y = -3x + 5
... y = -3x/2 + 5/2
Comparing
with the equation y = mx + c
...
the gradient m = -3/2 and the intercept
c = 5/2
...
the line passes through the point
( 0 , 5/2)
Substituting
y = 0
...
3x = 5
... x = 5/3
...
the line cuts the x axis at (5/3 ,
0)
Let
us now proceed to review vectors.
Please
review the following description:
(a)
A motor car travels with velocity
45Km per hour due north.
(b)
A force of 25 N due East.
Could
you say what both statements have
in common?
You
are correct that in both cases, their
sizes and directions are given. These
are examples of vector quantities
representing velocity of a car and
force, respectively.
A
vector quantity is one which identifies
both the magnitude (size) and direction,
for example, velocity given above.
A
speed of 20 metres per second is a
scalar quantity. (No direction given
Vector quantities are usually represented
in the form:
Vector
AB, a or as a column vector
(x/y)
We
will review Vectors represented as
Column vectors.
The
vector AB = (x/y) if x and y are respectively
the x and y components of the line
AB on the cartesian diagram.
EXAMPLE
Vector
AB = (4/3)
Please
Express
vector CD in the form (x/y)
If
your answer is (2/5) then you are
correct.
POINTS
TO NOTE
Avoid
making the common error of interchanging
x and y values.
If
the coordinates of A(x1
, y1) and B(x2
, y2) are given then the
vector AB = (x/y) where
x
= x2 - x1 and
y = y2 - y1
...
vector AB = (x/y) = ( (x2
- x1)/(y2 -
y1) )
you
may illustrate this on your own, using
the Cartesian diagram above.
(c)
A vector AB represented by (x/y) may
be placed anywhere in the Cartesian
diagram, as long as the x and y components
are satisfied. These would all be
equal vectors with the same length
and are in the smae direction and
are represented by (x/y)
(d)
If vector AB = (x/y) Then vector -AB
= (-x/-y) OR vector BA = (-x/-y)
It
is clear that the negative sign reverses
the direction of the vector.
Please
attempt the following for Homework:
EXAMPLE:
The vectors b, c and
d are given in the diagram
below. Express these in the form (p/q)
|
|
|
Manning's
School sixth-formers pay keen
attention during a motivational
session initiated by the Westmoreland
Chapter of the Manning's School
Past Students' Association,
recently.
- Photo by Dalton Laing
|
Clement
Radcliffe is the principal of Glenmuir
High School in May Pen.
|
