When something is moved around a fixed point, we say that it is
rotated about the point. What happens to the coordinates of a point that is rotated by
or
around the origin?
Investigation : rotation of a point by
Complete the table, by filling in the coordinates of the points shown in the figure.
Point
-coordinate
-coordinate
A
B
C
D
E
F
G
H
What do you notice about the
-coordinates? What do you notice about the
-coordinates?
What would happen to the coordinates of point A, if it was rotated to the position of point C? What about point B rotated to the position of D?
Investigation : rotation of a point by
Complete the table, by filling in the coordinates of the points shown in the figure.
Point
-coordinate
-coordinate
A
B
C
D
E
F
G
H
What do you notice about the
-coordinates? What do you notice about the
-coordinates?
What would happen to the coordinates of point A, if it was rotated to the position of point E? What about point F rotated to the position of B?
From these activities you should have come to the following conclusions:
90
clockwise rotation:
The image of a point P
rotated clockwise through 90
around the origin is P'
.
We write the rotation as
.
90
anticlockwise rotation:
The image of a point P
rotated anticlockwise through 90
around the origin is P'
.
We write the rotation as
.
180
rotation:
The image of a point P
rotated through 180
around the origin is P'
.
We write the rotation as
.
Rotation
For each of the following rotations about the origin:
(i) Write down the rule.(ii) Draw a diagram showing the direction of rotation.
OA is rotated to OA
with A(4;2) and A
(-2;4)
OB is rotated to OB
with B(-2;5) and B
(5;2)
OC is rotated to OC
with C(-1;-4) and C
(1;4)
Copy
XYZ onto squared paper. The co-ordinates are given on the picture.
Rotate
XYZ anti-clockwise through an angle of 90
about the origin to give
X
Y
Z
. Give the co-ordinates of X
, Y
and Z
.
Rotate
XYZ through 180
about the origin to give
X
Y
Z
. Give the co-ordinates of X
, Y
and Z
.
Enlargement of a polygon 1
When something is made larger, we say that it is
enlarged . What happens to the coordinates of a polygon that is enlarged by a factor
?
Investigation : enlargement of a polygon
Complete the table, by filling in the coordinates of the points shown in the figure.
Assume each small square on the plot is 1 unit.
Point
-coordinate
-coordinate
A
B
C
D
E
F
G
H
What do you notice about the
-coordinates? What do you notice about the
-coordinates?
What would happen to the coordinates of point A, if the square ABCD was enlarged by a factor 2?
Investigation : enlargement of a polygon 2
In the figure quadrilateral HIJK has been enlarged by a factor of 2 through the origin to become H'I'J'K'. Complete the following table using the information in the figure.
Co-ordinate
Co-ordinate'
Length
Length'
H = (;)
H' = (;)
OH =
OH' =
I = (;)
I' = (;)
OI =
OI' =
J = (;)
J' = (;)
OJ =
OJ' =
K = (;)
K' + (;)
OK =
OK' =
What conclusions can you draw about
the co-ordinates
the lengths when we enlarge by a factor of 2?
We conclude as follows:
Let the vertices of a triangle have co-ordinates S
, T
, U
.
S'T'U' is an enlargement through the origin of
STU by a factor of
(
).
STU is a reduction of
S'T'U' by a factor of
.
S'T'U' can alternatively be seen as an reduction through the origin of
STU by a factor of
. (Note that a reduction by
is the same as an enlargement by
).
The vertices of
S'T'U' are S'
, T'
, U'
.
The distances from the origin are OS' =
OS, OT' =
OT and OU' =
OU.
Transformations
Copy polygon STUV onto squared paper and then answer the following questions.
What are the co-ordinates of polygon STUV?
Enlarge the polygon through the origin by a constant factor of
. Draw this on the same grid. Label it S'T'U'V'.
What are the co-ordinates of the vertices of S'T'U'V'?
ABC is an enlargement of
A'B'C' by a constant factor of
through the origin.
What are the co-ordinates of the vertices of
ABC and
A'B'C'?
Giving reasons, calculate the value of
.
If the area of
ABC is
times the area of
A'B'C', what is
?
What are the co-ordinates of the vertices of polygon MNPQ?
Enlarge the polygon through the origin by using a constant factor of
, obtaining polygon M'N'P'Q'. Draw this on the same set of axes.
What are the co-ordinates of the new vertices?
Now draw M”N”P”Q” which is an anticlockwise rotation of MNPQ by 90
around the origin.
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .