6.3 Geometry  (Page 7/7)

Reflection on the line $y=x$

The final type of reflection you will learn about is the reflection of a point on the line $y=x$ .

Casestudy : reflection of a point on the line $y=x$

Study the information given and complete the following table:

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?

End of chapter exercises

1. 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)

      

2. 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.
3. 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.
   
4. 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)
5. 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.
6. 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.