<< Chapter < Page Chapter >> Page >

What can you deduce about the co-ordinates of points that are reflected about the line y = x ?

The x and y co-ordinates of points that are reflected on the line y = x are swapped around, or interchanged. This means that the x co-ordinate of the original point becomes the y co-ordinate of the reflected point and the y co-ordinate of the original point becomes the x co-ordinate of the reflected point.

Points A and B are reflected on the line y = x . The original points are shown with and the reflected points are shown with .
The x and y co-ordinates of points that are reflected on the line y = x are interchanged.

Find the co-ordinates of the reflection of the point R, if R is reflected on the line y = x . The co-ordinates of R are (-5;5).

  1. 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 y = x .

  2. The x co-ordinate of the reflected point is the y co-ordinate of the original point. Therefore, x =5.

    The y co-ordinate of the reflected point is the x co-ordinate of the original point. Therefore, y =-5.

  3. The co-ordinates of the reflected point are (5;-5).

Got questions? Get instant answers now!

Rules of Translation

A quick way to write a translation is to use a 'rule of translation'. For example ( x ; y ) ( x + a ; y + b ) means translate point (x;y) by moving a units horizontally and b units vertically.

So if we translate (1;2) by the rule ( x ; y ) ( x + 3 ; y - 1 ) 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.


Region A has been translated to region B by the rule: ( x ; y ) ( x + 4 ; y + 2 )

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
1. ( x ; y ) ( x ; y - 3 ) A. a reflection on x-y line
2. ( x ; y ) ( x - 3 ; y ) B. a reflection on the x axis
3. ( x ; y ) ( x ; - y ) C. a shift of 3 units left
4. ( x ; y ) ( - x ; y ) D. a shift of 3 units down
5. ( x ; y ) ( y ; x ) E. a reflection on the y-axis


  1. Describe the translations in each of the following using the rule (x;y) (...;...)
    1. From A to B
    2. From C to J
    3. From F to H
    4. From I to J
    5. From K to L
    6. From J to E
    7. From G to H
  2. 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.
    1. B is the reflection of A in the x-axis.
    2. C is the reflection of A in the y-axis.
    3. D is the reflection of B in the line x=0.
    4. E is the reflection of C is the line y=0.
    5. F is the reflection of A in the line y= x
  3. 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

  1. Look around the house or school and find a can or a tin of any kind (e.g. beans, soup, cooldrink, paint etc.)
  2. Measure the height of the tin and the diameter of its top or bottom.
  3. Write down the values you measured on the diagram below:
  4. Using your measurements, calculate the following (in cm 2 , rounded off to 2 decimal places):
    1. the area of the side of the tin (i.e. the rectangle)
    2. the area of the top and bottom of the tin (i.e. the circles)
    3. the total surface area of the tin
  5. If the tin metal costs 0,17 cents/cm 2 , how much does it cost to make the tin?
  6. Find the volume of your tin (in cm 3 , rounded off to 2 decimal places).
  7. What is the volume of the tin given on its label?
  8. 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.)
  9. Why do you think space is left for air in the tin?
  10. 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?
  11. If the height of the tin is kept the same, but now the radius is doubled, by what scale factor will the:
    1. area of the side surface of the tin increase?
    2. area of the bottom/top of the tin increase?


  • The properties of kites, rhombuses, parallelograms, squares, rectangles and trapeziums was covered. These figures are all known as quadrilaterals
  • You should know the formulae for surface area of rectangular and triangular prisms as well as cylinders
  • The volume of a right prism is calculated by multiplying the area of the base by the height. So, for a square prism of side length a and height h the volume is a × a × h = a 2 h .
  • Two polygons are similar if:
    • their corresponding angles are equal
    • the ratios of corresponding sides are equal
    . All squares are similar

End of chapter exercises

  1. Assess whether the following statements are true or false. If the statement is false, explain why:
    1. A trapezium is a quadrilateral with two pairs of parallel opposite sides.
    2. Both diagonals of a parallelogram bisect each other.
    3. A rectangle is a parallelogram that has all four corner angles equal to 60°.
    4. The four sides of a rhombus have different lengths.
    5. The diagonals of a kite intersect at right angles.
    6. Two polygons are similar if only their corresponding angles are equal.
  2. Calculate the area of each of the following shapes:
  3. Calculate the surface area and volume of each of the following objects (assume that all faces/surfaces are solid – e.g. surface area of cylinder will include circular areas at top and bottom):
  4. Calculate the surface area and volume of each of the following objects (assume that all faces/surfaces are solid):
  5. Using the rules given, identify the type of transformation and draw the image of the shapes.
    1. (x;y) (x+3;y-3)
    2. (x;y) (x-4;y)
    3. (x;y) (y;x)
    4. (x;y) (-x;-y)
  6. PQRS is a quadrilateral with points P(0; −3) ; Q(−2;5) ; R(3;2) and S(3;–2) in the Cartesian plane.
    1. Find the length of QR.
    2. Find the gradient of PS.
    3. Find the midpoint of PR.
    4. Is PQRS a parallelogram? Give reasons for your answer.
  7. 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.
  8. Consider: Triangle ABC with vertices A (1; 3) B (4; 1) and C (6; 4):
    1. Sketch triangle ABC on the Cartesian plane.
    2. Show that ABC is an isoceles triangle.
    3. Determine the co-ordinates of M, the midpoint of AC.
    4. Determine the gradient of AB.
    5. Show that the following points are collinear: A, B and D(7;-1)
  9. In the diagram, A is the point (-6;1) and B is the point (0;3)
    1. Find the equation of line AB
    2. Calculate the length of AB
    3. 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’
    4. Find the equation of A’B’
    5. Calculate the length of A’B’
    6. Can you state with certainty that AA'B'B is a parallelogram? Justify your answer.
  10. The vertices of triangle PQR have co-ordinates as shown in the diagram.
    1. 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.
    2. Determine the area of triangle PQR.
  11. Which of the following claims are true? Give a counter-example for those that are incorrect.
    1. All equilateral triangles are similar.
    2. All regular quadrilaterals are similar.
    3. In any A B C with A B C = 90 we have A B 3 + B C 3 = C A 3 .
    4. All right-angled isosceles triangles with perimeter 10 cm are congruent.
    5. All rectangles with the same area are similar.
  12. For each pair of figures state whether they are similar or not. Give reasons.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 10 maths [caps]' conversation and receive update notifications?