9.3 Transformations  (Page 3/3)

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.

Rules of Translation

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

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

Example

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

Discussion : rules of transformations

Work with a friend and decide which item from column 1 matches each description in column 2.

Transformations

1. Describe the translations in each of the following using the rule (x;y) $\to$ (...;...)

   

   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?

Summary

• 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 $a\times a\times h=a^{2}h$ .
• Similarity of polygons 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) $\to$ (x+3;y-3)

      

   2. (x;y) $\to$ (x-4;y)

      

   3. (x;y) $\to$ (y;x)

      

   4. (x;y) $\to$ (-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) $\to$ (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 $▵ABC$ with $\angle ABC={90}^{\circ }$ 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.