Spatial Sense

Rigid motions, such as translations and reflections, are fundamental ideas in geometry and transformations. Translation means shifting a shape without turning or changing its size, while reflection flips a shape over a line to create a mirror image. These transformations help in understanding symmetry and patterns in figures. Rotation, another type of rigid motion, involves turning a shape around a fixed point called the centre of rotation. This helps to analyse the orientation of shapes and predict their positions after rotation. Enlargement is a transformation that makes a shape bigger, or smaller, while keeping its form and proportions.

This is important for scaling objects in areas like architecture, engineering and art. Knowing how to enlarge shapes ensures accurate measurements and representations in these fields.

KEY IDEAS

• Enlargement: This involves increasing or decreasing the size of a shape by a certain scale factor while maintaining its proportions.

• Image: The object after a transformation is called the image. After a translation the image remains congruent and the orientation remains the same, but the position changes as determined by the translating vector.

After a reflection or a rotation, the images will be congruent to the original but will have been reflected or rotated from the original. After an enlargement the image will be similar to the original.

• Mirror line: A line over which the reflection occurs. Every point on the object has a corresponding point on the other side of the mirror line at an equal distance.

• Reflection: This is a transformation that “flips” an object across a mirror line to create a mirror image.

• Rotation: This is the circular movement of an object around a point. The angle of rotation, the centre of rotation and the direction (clockwise or counter clockwise) are key aspects.

• Rotational symmetry: An object or shape is said to have a rotational symmetry if it can be rotated about a central point and still look the same.

• Scale factor: A number that indicates how much bigger or smaller the image will be compared with the original shape.

• Transformation: This changes the size and/or the position of a shape.

There are four types of transformation: translation, reflection, rotation and enlargement

• Translating vector: This is a directed line segment with magnitude and direction that describes how far and in which direction each point of the object moves.

• Translation: A mathematical translation is a geometric transformation that moves a shape or object from one position to another on a plane without changing its size, shape or orientation.

TRANSFORMATIONS A transformation is a function that shifts or alters a shape in some manner to create a new shape known as the image. The original shape can also be referred to as the preimage. The points on the preimage serve as the inputs for the transformation, while the points on the image are the outputs. There are four types of transformation: translation, reflection, rotation and enlargement.

A vector is a quantity that has both direction and magnitude (size). On a coordinate plane, it is shown as an arrow that goes from one point to another.

A translation vector, often written as →v = ( a

b) shows how far and in which direction each point of a shape will move. For example, if →v = (− 1 3 ), it means every point on the shape is shifted one unit to the left and three units up. This means the shape stays the same in size and orientation, but it slides to a new position on the grid without turning or changing its size.

Rigid motion is a type of transformation where the size and shape of a figure remain unchanged. The only thing that changes is the figure’s position, but it does not alter in any other way.

In this section, you need to apply the knowledge gained in vectors from year 1.

Performing Translations: Study the diagram below.

Fig 3.1: The diagram shows a vector The diagram shows a vector. The starting point of the vector is S and the ending point is T. The vector is called ⟶ ST and is read as “vector ST.” The horizontal part of ⟶ ST is 5 and the vertical part is 3. When we combine the horizontal and vertical parts, we get what is called the component form of the vector. So, the component form of ST is (5 3).

Activity 3.1: Identifying translation vectors Study the graphs below and write down the components of given vectors.

Compare your answers with your classmates.

Figure 3.2 Figure 3.3

Figure 3.4

Translating a Figure Using a Vector: Study the following examples to enhance your understanding of translation.

Example 3.1

The vertices of △ABC are A (0, 3), B (2, 4) and C (1, 0). Translate △ABC using the vector ( 5 − 1)

Solution

First, graph △ABC. Use ( 5 − 1) to move each vertex 5 units right and 1 unit down.

Draw △A′B′C′, labelling the image vertices. Notice that the vectors drawn from preimage vertices to image vertices are parallel.

Figure 3.5: Graph of △ABC translated by ( 5 − 1) As a general rule, you can represent a translation along the vector ( a

b) using a rule, which is expressed as (x, y) → (x + a, y + b).

Applying this to example 1, to go from A to A′, you move 5 units right and 1 unit down, so you move along the vector ( 5 − 1) and the same applies for B to B’ and C to C’. So, a rule for the translation is (x, y) → (x + 5, y – 1).

Activity 3.2: Translating given coordinates using translation vectors The vertices of △PQR are P(1, 5), Q(3, 6), and R(2, 1). Translate △PQR using the vector ( 4 − 2). Write down the rule for the translation. Compare the results with your classmates.

Example 3.2

Graph quadrilateral ABCD with vertices A (−1, 2), B (−1, 5), C (4, 6), and D (4, 2) and its image after the translation (x, y) → (x + 3, y − 1).

Solution

Graph quadrilateral ABCD. To find the coordinates of the vertices of the image, add 3 to the x-coordinates and subtract 1 from the y-coordinates of the vertices of the preimage. Then graph the image, as shown.

(x, y) → (x + 3, y − 1)

Figure. 3.6: Graph of quadrilateral ABCD translated by (x, y) → (x + 3, y − 1) A (−1, 2) → A′ (2, 1) B (−1, 5) → B′ (2, 4) C (4, 6) → C′ (7, 5) D (4, 2) → D′ (7, 1)

Activity 3.3: Translating given coordinates using translation vectors The vertices of quadrilateral WXYZ are W (0, 2), X (3, 5), Y (5, 3), and Z (2, 0). Working in pairs, translate quadrilateral WXYZ using the vector ( 4 − 3).

Reflection is a transformation that uses a line as a “mirror” to reflect an image.

The diagram below illustrates reflection of images.

1. The “mirror” line, m, is called the Line of Reflection.

2. P and P′ are the same perpendicular distance from the line of reflection.

3. The line connecting P and P′ is perpendicular to the line of reflection.

Rules of Reflection

To perform a reflection in a coordinate plane, you need to follow some basic rules that relate to a given line of reflection.

Reflections across the x− Axis Rule: (x, y) → (x, −y) Description: When a point is reflected across the x-axis, the x-coordinate remains the same, while the y-coordinate changes sign.

Reflections across the y− Axis Rule: (x, y) → (−x, y) Description: When a point is reflected across the y-axis, the y-coordinate remains the same, while the x-coordinate changes sign.

Reflections across the line y = x Rule: (x, y) → (y, x) Description: When a point is reflected across the line y = x, the coordinates are swapped.

Reflections across the line y = −x Rule: (x, y) → (−y, −x) Description: When a point is reflected across the line y = −x, the coordinates are swapped and their signs are changed.

Reflections across a vertical line x = a Rule: (x, y) → (2a−x, y) Description: When a point is reflected across a vertical line x = a, the y-coordinate remains the same, while the x-coordinate is transformed by the formula 2a − x.

Reflections across a horizontal line y = b Rule: (x, y) → (x, 2b−y) Description: When a point is reflected across a horizontal line y = b, the x-coordinate remains the same, while the y-coordinate is transformed by the formula 2b − y.

Example 3.3

The vertices of ΔLMN are L (−3, 3), M (1, 2), and N (−2, 1). Plot the coordinates on a graph and join the points to form ΔLMN. Draw the image ΔL’M’N’ of ΔLMN using the y-axis as the mirror line.

Solution

Figure 3.7: Graph of ΔLMN reflected in the y-axis From the graph, we can identify the coordinates as;

(a, b) → (−a, b) L (−3, 3) → L′(3, 3) M (1, 2) → M′(−1, 2) N(−2, 1) → N′(2, 1)

Activity 3.4: Identifying reflections of objects in a mirror line Working in pairs, study the diagrams below and describe the type of reflections in each. Remember that all reflections must be defined by their ‘mirror line’, so be sure to state correctly the line of reflection.

Figure 3.8: Graph of a reflected object

Figure 3.9: Graph of a reflected object

Activity 3.5: Reflecting objects on a mirror line Using the diagrams below reflect ΔABC in the x axis and parallelogram ABCD in the y axis. Compare your answers with your classmates.

Figure 3.10: Reflecting objects on a mirror line

Example 3.4

Graph ΔABC with vertices A (1, 3), B (4, 4), and C (3, 1). Reflect ΔABC in the line y = −x.

Figure 3.11: Graph of ΔABC in the line y = −x From the graph, we can see how ΔABC is reflected in the mirror line y = −x.

Study the dash lines carefully to understand how each point is reflected.

Rotation of Images of Plane Shape

A rotation is when a shape is turned around a fixed point, called the centre of rotation, by a certain angle and direction, usually measured in degrees. This movement keeps the shape the same size and proportion but changes its orientation. During a rotation, each point on the shape follows a circular path around the centre. The angle of rotation shows how far the shape turns, and the direction (clockwise or anti-clockwise) dictates the direction in which it moves. Common rotation angles are 90°, 180°, and 270°, though any angle can be used.

Real-life Examples of Rotation

In real life, we see rotation in things like a spinning wheel, a ceiling fan, or the hands of a clock. Below are some examples you may find around you

Figure 3.12: Real-life Examples of Rotation

Rotations can be illustrated using graphs as illustrated below.

Figure 3.13: Rotation in a coordinate plane

Activity 3.6: Investigating real-life examples of rotation Working in small groups, look for objects around the school or outside the school that demonstrate rotation. Write the list of those objects and describe how they rotate (clockwise or counter clockwise) to your classmates. If possible, take videos of how these objects rotate and share these.

Coordinate Rules for Counterclockwise

(anticlockwise) Rotations about the Origin Study the diagrams below to see how rotations of 90⁰, 180⁰, 270⁰, and 360⁰about the origin are done.

Figure 3.14: Rotations 90°, 180°, 270°, and 360° about the origin From the above diagram, you can make the following conclusions:

90° Anticlockwise Rotation (or 270° Clockwise Rotation)

Rule: (x, y) → (−y, x) Description: When a point is rotated 90° anticlockwise about the origin, the x-coordinate becomes the negative y-coordinate, and the y-coordinate becomes the x-coordinate.

180° Rotation (Anticlockwise or Clockwise)

Rule: (x, y) → (−x, −y) Description: When a point is rotated 180° about the origin, both the x- and y-coordinates change signs.

270° Anticlockwise Rotation (or 90° Clockwise Rotation)

Rule: (x, y) → (y, −x) Description: When a point is rotated 270° anticlockwise (or 90° clockwise) about the origin, the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative x-coordinate.

Example 3.5

If E (−3, 2), F (−3, 4), G (1, 4), and H (2, 2). Find the image matrix for a 270° anticlockwise rotation about the origin.

Solution

Figure 3.15: Graph of EFGH rotated 270° anticlockwise about the origin (a, b) → (b, −a) E(−3, 2) → E′(2, 3) F(−3, 4) → F′(4, 3) G(1, 4) → G′(4, −1) H(2, 2) → H′(2, −2)

Example 3.6

Graph △JKL with vertices J(3, 0), K(4,3) L(6,0) and its image after a 90° clockwise rotation about the origin.

Figure 3.16: Graph of △JKL 90° through clockwise rotation about the origin

Example 3.7

Graph △ABC with vertices A(0, 3), B(-4,0) and C (0, -1) and its image after a 270° clockwise rotation about the point P(2,0).

Solution

Remember that a 270° clockwise rotation is the same as a 90° anti clockwise rotation.

Figure 3.17: Graph of △ABC with 270° clockwise rotation about the point P (2,0)

Activity 3.7: Rotation of objects in a coordinate plane By looking at the graph below, describe how to locate point A′ if point A is rotated 270⁰clockwise around the origin.

Figure 3.18: Locating point A′ if point A is rotated 270° clockwise around the origin

Activity 3.8: Rotation of objects in a coordinate plane Using the graph below

1. Rotate point A 90⁰counterclockwise around the origin (0, 0). Plot the new position and label it A′.

2. Repeat the rotation of A by 180⁰and then 270⁰counterclockwise around the origin. Plot these points and label them A′′ and A′′′ respectively.

3. Record the coordinates of each point after rotation.

4. Describe any patterns you observe in the coordinates after each rotation.

Figure. 3.19: Locating the image of A Rotational Symmetry: A figure has rotational symmetry if it looks the same after being rotated a certain angle. Examples of some objects that exhibit rotational symmetry are: Gye Nyame, Fawohodie, and Eban

Activity 3.9: Exploring Rotational Symmetry with Adinkra Symbols

Materials Needed

• Printed sheets of various Adinkra symbols (examples, Gye Nyame, Fawohodie, Eban)

• Scissors

• Markers

• Blank paper for drawing Steps:

1. Cut out an Adinkra symbol, for example, Gye nyame (representing the supremacy of God), or Fawohodie (symbolising independence) or Eban (symbolising security and safety) or any other Ghanaian symbol.

2. Place the symbol on a blank sheet of paper.

3. Rotate the symbol by 90°, 180° and 270° and mark each position with a light pencil outline.

4. Note the angle(s) that make(s) the symbol look the same as the original position.

5. Share your observations with your classmates. Discuss with them, why some symbols look the same at multiple rotations while others only match after a full 360° turn.

6. Connect the idea of rotational symmetry to Ghanaian art and culture.

7. Discuss with your classmates how symmetry adds beauty and balance in design.

Activity 3.10: Exploring rotational symmetry Work in small groups, in pairs or individually to create your own symbols or patterns that have rotational symmetry. Show them to your classmates or teacher and explain the rotational symmetry incorporated in each symbol or pattern.

Activity 3.11: Rotational symmetry of a geometric figure The figure below shows the rotational symmetry of a rectangle. By looking at the illustrations, identify a rectangular object in your immediate environment and discuss its rotational symmetry with your classmates.

Figure 3.20: Order of rotational symmetry The above example shows the rotation of a rectangle 90° each time. The rectangle has the rotational symmetry of order 2 because when it is rotated twice, we get the original shape at 180° and again, when it is rotated twice, the original form is obtained at 360°. So, the order of rotational symmetry of the rectangle is 2.

Enlargement of Plane Shapes

Enlargement (or dilation) is a type of transformation that changes the size of a shape while maintaining its overall geometry. Enlargement can either increase the size of a shape (magnification) or decrease it (reduction), depending on the scale factor. Unlike other rigid motions such as translation, rotation and reflection, which maintain the size and shape of the figure, enlargement changes the size but keeps the shape similar.

In an enlargement, a plane shape is scaled by a specific factor relative to a fixed point called the centre of enlargement. This scale factor, often denoted as “k”, determines how much the shape is enlarged or reduced. If k > 1, the shape is enlarged; if k < 1, the shape is reduced in size. The transformation ensures that all points on the original shape (the pre-image) move radially outward or inward from the centre of enlargement by a distance proportional to k.

The mathematical rules governing enlargement involve multiplying the coordinates of each point in the original shape by the scale factor k, resulting in a new shape (the image) that is similar to the original but proportionally larger or smaller.

Look at the diagram:

Figure 3.21: Enlargement of Shape A

The example shows how the original, A, was enlarged with scale factors 2 and 4.

A line from the centre of enlargement passes through the corresponding vertex of each image.

Note

The distances, OA′ and OA′′, are related to OA by:

OA′ = 2 × OA OA′′ = 4 × OA The same is true of all the other distances between O and corresponding points on the images.

Activity 3.12: Enlarging plane shapes using graph paper Materials Needed

• Graph paper

• Ruler

• Pencil

• Coloured markers Steps

1. Draw a triangle or rectangle, on graph paper. Ensure the shape has coordinates that are easy to identify by placing vertices at grid intersections.

2. Identify the centre of enlargement at the origin (0,0).

3. Multiply each coordinate of the original shape by the scale factor to get the new enlarged coordinates. For example, if a vertex is at (2, 3) in the original shape and the scale factor is 2, the new coordinate would be (4,

6) when the centre of enlargement is at the origin.

4. Plot these new coordinates and join them to form the enlarged shape.

5. Measure the sides of the original and enlarged shapes. You will observe that each side of the enlarged shape is exactly twice (or the scale factor times) the corresponding side of the original shape.

6. Discuss with your classmates, how the angles remain the same and that the shapes are similar.

7. Experiment with different scale factors (e.g., 3, 0.5) to see how the shape changes with enlargement and reduction.

Example 3.8

Plot the coordinates A (2, 1) B (1, 3) C (3, 2) of △ABC. Perform an enlargement of △ABC using a scale factor of 2 and a centre of the enlargement at the origin to form △A′B′C′.

Compare the coordinates, side lengths and angle measures of △ABC and △A′B′C′.

Figure 3.22: Enlargement of △ABC Sometimes, the scale factor is a negative number. When this happens, the figure rotates 180°. So, when k > 0, a dilation with a scale factor of −k is the same as the composition of a dilation with a scale factor of k followed by a rotation of 180° about the centre of dilation. Using the coordinate rules for a dilation centred at (0, 0) and a rotation of 180°, you can think of the notation as (x, y) → (kx, ky) → (−kx, −ky).

Example 3.9

Graph △FGH with vertices F(−4, −2), G(−2, 4), and H(−2, −2) and its image after a dilation centred at (0, 0) with a scale factor of − 1/2 .

Solution

Use the coordinate rule for a dilation with centre (0, 0) and k = –1/2 to find the coordinates of the vertices of the image.

Figure 3.23: Graph of △FGH dilated at centre (0, 0) and scale factor − 1/2 .

Then graph △FGH and its image.

(x, y) → (−1/2 x, − 1/2 y) F(−4, −2) → F′(2, 1) G(−2, 4) → G′(1, −2) H(−2, −2) → H′(1, 1)

Example 3.10

Graph △FGH with vertices F(−4, −2), G(−2, 4), and H(−2, −2) and its image after a dilation centred at (0, 0) with a scale factor of 1/2 .

Figure 3.24: Graph of △FGH and its image Graph △FGH and its image.

(x, y) → (1/2 x, 1__ 2y) F(−4, −2) → F′(-2, -1) G(−2, 4) → G′(-1, 2) H(−2, −2) → H′(-1,-1) Looking at Examples 2 and 3, it should be noted that the scale factor had different effects on the image of △FGH. The negative scale factor is affected differently from the positive scale factor as it shows a rotation about the origin as well as the dilation.

Applications

Example 3.11

You are using a magnifying glass that shows the image of an object that is six times the object’s actual size. Determine the length of the image of the spider seen through the magnifying glass.

Solution

image length_ actual length = k x_ 1.5 = 6 x = 9 The image length through the magnifying glass is 9 centimetres.

Example 3.12

In Accra, a construction company is building a new community park with a traditional Ghanaian Kente-inspired mosaic on the main walkway. The original Kente design on a cloth measures 50cm × 80cm. To make it suitable for the walkway, the company needs to enlarge the design by a scale factor of 5.

1. Calculate the dimensions of the enlarged Kente mosaic that will be laid on the walkway.

2. Find the area of the original design and the area of the enlarged mosaic.

3. How many times greater is the area of the enlarged mosaic compared with the original design?

4. If the cost of laying the mosaic is GH¢20 per square metre, calculate the total cost to lay the enlarged Kente mosaic on the walkway.

Solution

1. Original dimensions = 50cm × 80cm Enlarged dimensions = 50cm × 5 = 250cm = 80cm × 5 = 400cm Dimensions of enlarged mosaic = 250cm × 400cm

2. Original length in metres = 0.5m Original width in metres = 0.8m Original area = 0.5m × 0.8m = 0.4m²New length in metres = 2.5m New width in metres = 4 m Enlarged Area = 2.5m × 4m = 10m²3. Scale factor, k = Area of enlarged mosaic__________________ Area of original mosaic = 10m²_____ 0.4m²= 25 (Note that this is the length scale factor squared)

4. Total cost to lay the enlarged Kente mosaic on the walkway = Area of enlarged mosaic × GH¢20 =10m²× GH¢20 = GH¢200.00.

1. If a point is at A(1, 2) and is translated 3 units to the right, what is the new coordinate of A′?

2. What is the coordinates of the point (3, 4) after it is reflected across the x-axis?

3. A triangle has vertices at A(2, 3), B(4, 5), and C(6, 2). If the triangle is translated by the vector v = (− 3 4 ), what are the new coordinates of the vertices A′, B′, and C′?

4. Reflect the shape with vertices P(1, 2), Q(3, 5) and R(2, -1) across the line x = 0 (the y-axis). What are the new coordinates of the reflected vertices?

5. A quadrilateral has vertices at A(1, 2), B(3, 5), C(5, 2), and D(3, 0). After a translation by the vector v = (− 2 3 ), the quadrilateral is mapped onto another position. Determine the coordinates of the new vertices and explain how the shape and orientation of the quadrilateral are affected by the translation.

6. A rectangle has vertices at A(-3, 1), B(-1, 4), C(2, 4), and D(0, 1). Reflect the rectangle across the line y = 2 and find the coordinates of the new vertices. Explain how the position of the shape changes after the reflection.

7. Given a triangle with vertices A(1, 3), B(4, 7), and C(6, 2), it undergoes two successive translations: first by the vector v₁= ( 2 − 3), and second by the vector v₂= (− 5 4 ).

Find the final coordinates of the triangle and explain how the composition of these translations affects the overall movement.

8. Consider a parallelogram with vertices P(-2, 1), Q(-1, 4), R(3, 4), and S(2, 1). Reflect the parallelogram across the line, y = -1, and then reflect the resulting image across the y-axis. What are the coordinates of the final vertices after both reflections? Analyse how these two reflections combine to transform the shape.

9. Graph the line RS with points R(1, −3) and S(2, −6). Rotate the line through180° about the origin. Then reflect RS in the y-axis.

10. Graph ∆ABC with vertices A(−4, 4), B(−1, 7), and C(-1,4) and its image after a 270⁰clockwise rotation about the origin.

11. Graph △PQR and its image after an enlargement centred at C with scale factor k.

a. P(−2, −1), Q(−1, 0), R(0, −1); C(0, 0), k = 4

b. P(5, −5), Q(10, −5), R(10, 5); C(0, 0), k = 0.4

12. The image of a spider seen through the magnifying glass is 4 times its actual length as shown in the diagram.

Find the actual length of the spider.