When a point is reflected on the
-axis, only the
co-ordinate of the point changes. The
co-ordinate remains unchanged.
Find the co-ordinates of the reflection of the point Q, if Q is reflected on the
-axis. The co-ordinates of Q are (15;5).
We are given the point Q with co-ordinates (15;5) and need to find the co-ordinates of the point if it is reflected on the
-axis.
The point Q is to the right of the
-axis, therefore its reflection will be the same distance to the left of the
-axis as the point Q is to the right of the
-axis. Therefore,
=-15.
For a reflection on the
-axis, the
co-ordinate remains unchanged. Therefore,
=5.
The co-ordinates of the reflected point are (-15;5).
Reflection on the line
The final type of reflection you will learn about is the reflection of a point on the line
.
Casestudy : reflection of a point on the line
Study the information given and complete the following table:
Point
Reflection
A
(2;1)
(1;2)
B
(-
;-2)
(-2;-1
)
C
(-1;1)
D
(2;-3)
What can you deduce about the co-ordinates of points that are reflected about the line
?
The
and
co-ordinates of points that are reflected on the line
are swapped around, or interchanged. This means that the
co-ordinate of the original point becomes the
co-ordinate of the reflected point and the
co-ordinate of the original point becomes the
co-ordinate of the reflected point.
The
and
co-ordinates of points that are reflected on the line
are interchanged.
Find the co-ordinates of the reflection of the point R, if R is reflected on the line
. The co-ordinates of R are (-5;5).
We are given the point R with co-ordinates (-5;5) and need to find the co-ordinates of the point if it is reflected on the line
.
The
co-ordinate of the reflected point is the
co-ordinate of the original point. Therefore,
=5.
The
co-ordinate of the reflected point is the
co-ordinate of the original point. Therefore,
=-5.
The co-ordinates of the reflected point are (5;-5).
Rules of Translation
A quick way to write a translation is to use a 'rule of translation'. For example
means translate point (x;y) by moving a units horizontally and b units vertically.
So if we translate (1;2) by the rule
it becomes (4;1). We have moved 3 units right and 1 unit down.
Translating a Region
To translate a region, we translate each point in the region.
Example
Region A has been translated to region B by the rule:
Discussion : rules of transformations
Work with a friend and decide which item from column 1 matches each description in column 2.
Column 1
Column 2
a reflection on x-y line
a reflection on the x axis
a shift of 3 units left
a shift of 3 units down
a reflection on the y-axis
Transformations
Describe the translations in each of the following using the rule (x;y)
(...;...)
From A to B
From C to J
From F to H
From I to J
From K to L
From J to E
From G to H
A is the point (4;1). Plot each of the following points under the given transformations. Give the co-ordinates of the points you have plotted.
B is the reflection of A in the x-axis.
C is the reflection of A in the y-axis.
D is the reflection of B in the line x=0.
E is the reflection of C is the line y=0.
F is the reflection of A in the line y= x
In the diagram, B, C and D are images of polygon A. In each case, the transformation that has been applied to obtain the image involves a reflection and a translation of A. Write down the letter of each image and describe the transformation applied to A in order to obtain the image.
Investigation : calculation of volume, surface area and scale factors of objects
Look around the house or school and find a can or a tin of any kind (e.g. beans, soup, cooldrink, paint etc.)
Measure the height of the tin and the diameter of its top or bottom.
Write down the values you measured on the diagram below:
Using your measurements, calculate the following (in cm
, rounded off to 2 decimal places):
the area of the side of the tin (i.e. the rectangle)
the area of the top and bottom of the tin (i.e. the circles)
the total surface area of the tin
If the tin metal costs 0,17 cents/cm
, how much does it cost to make the tin?
Find the volume of your tin (in cm
, rounded off to 2 decimal places).
What is the volume of the tin given on its label?
Compare the volume you calculated with the value given on the label. How much air is contained in the tin when it contains the product (i.e. cooldrink, soup etc.)
Why do you think space is left for air in the tin?
If you wanted to double the volume of the tin, but keep the radius the same, by how much would you need to increase the height?
If the height of the tin is kept the same, but now the radius is doubled, by what scale factor will the:
area of the side surface of the tin increase?
area of the bottom/top of the tin increase?
End of chapter exercises
Using the rules given, identify the type of transformation and draw the image of the shapes.
(x;y)
(x+3;y-3)
(x;y)
(x-4;y)
(x;y)
(y;x)
(x;y)
(-x;-y)
PQRS is a quadrilateral with points P(0; −3) ; Q(−2;5) ; R(3;2) and S(3;–2) in the Cartesian plane.
Find the length of QR.
Find the gradient of PS.
Find the midpoint of PR.
Is PQRS a parallelogram? Give reasons for your answer.
A(–2;3) and B(2;6) are points in the Cartesian plane. C(a;b) is the midpoint of AB. Find the values of a and b.
Consider: Triangle ABC with vertices A (1; 3) B (4; 1) and C (6; 4):
Sketch triangle ABC on the Cartesian plane.
Show that ABC is an isoceles triangle.
Determine the co-ordinates of M, the midpoint of AC.
Determine the gradient of AB.
Show that the following points are collinear: A, B and D(7;-1)
In the diagram, A is the point (-6;1) and B is the point (0;3)
Find the equation of line AB
Calculate the length of AB
A’ is the image of A and B’ is the image of B. Both these images are obtain by applying the transformation: (x;y)
(x-4;y-1). Give the coordinates of both A’ and B’
Find the equation of A’B’
Calculate the length of A’B’
Can you state with certainty that AA'B'B is a parallelogram? Justify your answer.
The vertices of triangle PQR have co-ordinates as shown in the diagram.
Give the co-ordinates of P', Q' and R', the images of P, Q and R when P, Q and R are reflected in the line y=x.
Determine the area of triangle PQR.
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?