Mathwise Academy | Grade 8 Test 1 - Geometry of 2D Shapes and Straight Lines
M

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Geometry of 2D Shapes and Straight Lines

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

  • Answer ALL the questions.
  • Read all questions carefully before you begin.
  • Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
  • You MUST provide valid GEOMETRIC REASONS for all statements made when calculating unknown angles or proving geometric properties.
  • Write neatly and legibly.

QUESTION 1: PROPERTIES OF TRIANGLES AND QUADRILATERALS [12]

1.1 Classify the following triangles according to their angles and sides. State both classifications for each triangle:

  • 1.1.1 A triangle with angles \( 40^\circ \), \( 60^\circ \), and \( 80^\circ \). (2)
  • 1.1.2 A triangle with sides \( 8\text{ cm} \), \( 8\text{ cm} \), and \( 12\text{ cm} \), and one angle measuring \( 110^\circ \). (2)

1.2 In a parallelogram \( PQRS \), \( \hat{P} = 115^\circ \). Write down the sizes of \( \hat{Q} \), \( \hat{R} \), and \( \hat{S} \). Provide brief geometric reasons for your answers.

[4]

1.3 State whether the following statements are TRUE or FALSE. If false, briefly explain why:

  • 1.3.1 All rhombuses are squares. (2)
  • 1.3.2 The diagonals of a rectangle always bisect each other at \( 90^\circ \). (2)

QUESTION 2: CALCULATING UNKNOWN ANGLES IN 2D SHAPES [18]

2.1 In \( \triangle DEF \), the interior angles are given in terms of \( x \):
\( \hat{D} = 2x + 10^\circ \), \( \hat{E} = 3x - 20^\circ \) and \( \hat{F} = x + 40^\circ \).

D E F 2x + 10° 3x - 20° x + 40°
  • 2.1.1 Calculate the value of \( x \), with reasons. (4)
  • 2.1.2 Determine the actual size of the largest angle in the triangle. (2)

2.2 Consider the diagram below. \( \triangle KLM \) is an isosceles triangle with \( KL = KM \). The exterior angle at \( M \) is \( \hat{M}_2 = 138^\circ \).

K L M N 1 2 138°

Calculate the sizes of the interior angles \( \hat{M}_1 \), \( \hat{L} \), and \( \hat{K} \). Show all steps and provide reasons.

[6]

2.3 In quadrilateral \( ABCD \), it is given that \( \hat{A} = 2x \), \( \hat{B} = 3x - 10^\circ \), \( \hat{C} = x + 30^\circ \) and \( \hat{D} = 4x \).

A B C D 2x 3x - 10° x + 30° 4x
  • 2.3.1 Calculate the value of \( x \). (4)
  • 2.3.2 Calculate the size of angle \( \hat{D} \). (2)

QUESTION 3: GEOMETRY OF STRAIGHT LINES (HIGH ORDER) [20]

3.1 Two straight lines, \( AB \) and \( CD \), intersect at \( E \). The angle \( A\hat{E}C \) is \( 5x - 40^\circ \) and the vertically opposite angle \( D\hat{E}B \) is \( 3x + 20^\circ \).

A B C D E 5x - 40° 3x + 20° AÊD = ?
  • 3.1.1 Calculate the value of \( x \), with reasons. (3)
  • 3.1.2 Calculate the size of angle \( A\hat{E}D \). (3)

3.2 In the diagram below, line \( PQ \) is parallel to line \( RS \) (\( PQ \parallel RS \)). The transversal line \( TU \) cuts \( PQ \) at \( V \) and \( RS \) at \( W \). Angle \( P\hat{V}W = 3x - 15^\circ \) and angle \( V\hat{W}S = 2x + 45^\circ \).

P Q R S T U V W 3x - 15° 2x + 45°
  • 3.2.1 Identify the relationship between \( P\hat{V}W \) and \( V\hat{W}S \). (1)
  • 3.2.2 Formulate an equation and calculate the value of \( x \), providing the necessary geometric reason. (4)
  • 3.2.3 Hence, determine the size of the alternate angle to \( V\hat{W}S \). (2)

3.3 Complex Straight Line Problem:
In the figure below, \( AB \parallel CD \). Triangle \( \triangle EFG \) lies between the parallel lines with vertex \( F \) on line \( AB \) and vertex \( G \) on line \( CD \). You are given that \( A\hat{F}E = 48^\circ \) and \( E\hat{G}D = 62^\circ \). Furthermore, \( \triangle EFG \) is an isosceles triangle with \( EF = EG \).

A B C D F E G 48° 62° FÊG = ?

3.3.1 Calculate the size of the interior angle \( F\hat{E}G \) showing all logical geometric steps and providing reasons.
(Hint: Draw an imaginary construction line through \( E \) parallel to \( AB \) and \( CD \)).

[7]
--- END OF PAPER ---
Mathwise Academy | Grade 8 Test 2 - Geometry of 2D Shapes and Straight Lines (Higher Order)
M

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Geometry of 2D Shapes and Straight Lines (Higher Order)

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

  • Answer ALL the questions.
  • Read all questions carefully before you begin.
  • Show ALL working out. Full marks will not necessarily be awarded for answers only.
  • You MUST provide valid GEOMETRIC REASONS for every statement you make when calculating unknown angles or sides.
  • Write neatly and legibly.

QUESTION 1: ALGEBRA IN 2D SHAPES AND SIMILARITY [15]

1.1 In parallelogram \( ABCD \) below, the size of interior angle \( \hat{A} \) is given as \( 3x - 10^\circ \) and the size of the interior opposite angle \( \hat{C} \) is given as \( 2x + 20^\circ \).

A D C B 3x - 10° 2x + 20° y
  • 1.1.1 Set up an equation and calculate the value of \( x \), giving a reason for your statement. (3)
  • 1.1.2 If the size of interior angle \( \hat{B} \) is given as \( y \), calculate the value of \( y \). Give a valid reason. (3)

1.2 In the diagram below, \( PQ \) is parallel to \( ST \) (\( PQ \parallel ST \)). The line segments \( PT \) and \( QS \) intersect at point \( R \). The lengths of the sides are given as follows: \( PQ = 15\text{ cm} \), \( ST = 5\text{ cm} \), and \( PR = 12\text{ cm} \).

P Q R S T PQ = 15 cm ST = 5 cm PR = 12 cm TR = ?
  • 1.2.1 Prove, with valid reasons, that \( \triangle PQR \) is similar to \( \triangle TSR \) (\( \triangle PQR \|| \triangle TSR \)). (4)
  • 1.2.2 Hence, use the properties of similar triangles to calculate the length of side \( TR \). (5)

QUESTION 2: PROVING PROPERTIES OF STRAIGHT LINES [12]

2.1 Two straight lines intersect at point \( M \), forming four angles around the point. Two of these vertically opposite angles are given as \( 4x + 10^\circ \) and \( 5x - 20^\circ \).

M 4x + 10° 5x - 20°
  • 2.1.1 Calculate the value of \( x \). (3)
  • 2.1.2 Hence, calculate the size of the obtuse angles formed at the intersection. (2)

2.2 In the diagram below, line \( AB \) and line \( CD \) are cut by transversal line \( EF \) at points \( G \) and \( H \) respectively. You are given the interior angles: \( A\hat{G}H = 3x + 20^\circ \), \( D\hat{H}G = x \), and \( C\hat{H}G = 140^\circ \).

A B C D E F G H 3x + 20° 140° x
  • 2.2.1 Calculate the value of \( x \) using the angles on the straight line \( CD \). (3)
  • 2.2.2 Hence, prove with calculations and a reason that line \( AB \) is parallel to line \( CD \) (\( AB \parallel CD \)). (4)

QUESTION 3: GEOMETRY OF STRAIGHT LINES - HIGH ORDER [23]

3.1 (The Zig-Zag Problem)
In the diagram below, line \( PQ \) is parallel to line \( RS \) (\( PQ \parallel RS \)). Two line segments connect points \( P \) and \( R \) to a central point \( V \). You are given that \( Q\hat{P}V = 42^\circ \) and \( S\hat{R}V = 58^\circ \).

P Q R S V 42° 58° PVR = ?
  • 3.1.1 Calculate the size of the interior angle \( P\hat{V}R \).
    (Hint: Draw an imaginary line passing through \( V \) that is parallel to \( PQ \) and \( RS \)).     (5)
  • 3.1.2 Hence, write down the size of the reflex angle \( P\hat{V}R \). (2)

3.2 In the diagram below, parallel lines \( AB \) and \( CD \) are intersected by a transversal \( EF \) at points \( G \) and \( H \). The corresponding angles are given as expressions in terms of \( x \):
\( E\hat{G}B = 4x - 10^\circ \) and \( G\hat{H}D = 2x + 50^\circ \). Another angle, vertically opposite to \( E\hat{G}B \), is labeled \( y \).

A B C D E F G H 4x - 10° y 2x + 50°
  • 3.2.1 Formulate an equation and calculate the value of \( x \), with a valid reason. (4)
  • 3.2.2 Calculate the size of angle \( y \), providing a geometric reason for your step(s). (4)

3.3 In the diagram below, line \( LM \) is parallel to line \( NO \) (\( LM \parallel NO \)). An isosceles triangle \( \triangle VWX \) is positioned such that point \( V \) lies on line \( LM \), and points \( W \) and \( X \) lie on line \( NO \).
You are given that \( VW = VX \) (indicated by the tick marks) and the alternate angle \( M\hat{V}X = 64^\circ \).

L M N O V W X 64°
  • 3.3.1 Write down the size of \( V\hat{X}W \), giving a valid reason. (2)
  • 3.3.2 State the size of \( V\hat{W}X \), giving a valid reason. (2)
  • 3.3.3 Hence, calculate the size of \( W\hat{V}X \) inside the triangle, providing your reason. (4)
--- END OF PAPER ---
Mathwise Academy | Grade 8 Test 3 - Geometry of 2D Shapes and Straight Lines (Additional Applications)
M

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Geometry of 2D Shapes and Straight Lines (Additional Applications)

Time: 1 Hour

Total Marks: 47

INSTRUCTIONS AND INFORMATION

  • Answer ALL the questions.
  • Read all questions carefully before you begin.
  • Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
  • You MUST provide valid GEOMETRIC REASONS for every statement you make when calculating unknown angles or sides.
  • Write neatly and legibly.

QUESTION 1: PROPERTIES OF 2D SHAPES [15]

1.1 Classify the following triangles according to their sides. Choose from: Scalene, Isosceles, or Equilateral.

  • 1.1.1 A triangle with sides \( 5\text{ cm} \), \( 8\text{ cm} \), and \( 5\text{ cm} \). (1)
  • 1.1.2 A triangle with angles \( 60^\circ \), \( 60^\circ \), and \( 60^\circ \). (1)
  • 1.1.3 A triangle with sides \( 4\text{ cm} \), \( 6\text{ cm} \), and \( 9\text{ cm} \). (1)

1.2 Name the specific type of quadrilateral described by each set of properties below:

  • 1.2.1 A quadrilateral with only one pair of opposite sides parallel. (1)
  • 1.2.2 A parallelogram with four equal sides and four right angles. (1)
  • 1.2.3 A quadrilateral with two pairs of adjacent sides equal. (1)

1.3 In the diagram below, \( ABCD \) is a rectangle. The length of \( AB \) is \( 8\text{ cm} \) and \( AD \) is \( 5\text{ cm} \).

A B C D 8 cm 5 cm
  • 1.3.1 Write down the length of side \( BC \). (1)
  • 1.3.2 Write down the size of angle \( \hat{B} \). (1)
  • 1.3.3 Calculate the perimeter of rectangle \( ABCD \). (2)

1.4 In \( \triangle PQR \) below, the sides \( PQ \) and \( PR \) are equal. The angle at vertex \( P \) is \( 70^\circ \).

P Q R 70° x y
  • 1.4.1 What type of triangle is \( \triangle PQR \)? (1)
  • 1.4.2 Calculate the size of angle \( x \), showing your working and giving reasons. (4)

QUESTION 2: GEOMETRY OF STRAIGHT LINES (ANGLES) [15]

2.1 Define the following angle terms:

  • 2.1.1 Complementary angles. (1)
  • 2.1.2 Supplementary angles. (1)

2.2 Calculate the size of the unknown angle \( x \) in the diagram below. Provide a reason for your answer.

A B C D 65° x
[3]

2.3 Two straight lines intersect at point \( M \), forming vertically opposite angles as shown in the diagram.

M 110° y x
  • 2.3.1 Write down the value of \( y \), with a reason. (2)
  • 2.3.2 Calculate the value of \( x \), with a reason. (3)

2.4 Determine the value of \( a \) in the diagram below, showing your working and reason.

40° a 60°
[5]

QUESTION 3: PARALLEL LINES [17]

3.1 In the diagram below, line \( AB \) is parallel to line \( CD \) (\( AB \parallel CD \)). Transversal line \( EF \) intersects the parallel lines. Calculate the sizes of angles \( x \), \( y \), and \( z \). You must provide a valid geometric reason for each answer.

A B C D E F 115° x y z
  • 3.1.1 Calculate \( x \). (3)
  • 3.1.2 Calculate \( y \). (3)
  • 3.1.3 Calculate \( z \). (3)

3.2 In the diagram below, \( PQ \parallel RS \). The transversal line creates an alternate angle to \( 55^\circ \) marked as \( a \), and a co-interior angle marked as \( b \).

P Q R S 55° b a
  • 3.2.1 Write down the value of \( a \), with a reason. (2)
  • 3.2.2 Calculate the value of \( b \), with a reason. (3)

3.3 Identifying Parallel Lines:
In the diagram below, you are given two straight lines \( LM \) and \( NO \) cut by transversal \( PQ \). The interior angles on the same side of the transversal are given as \( 72^\circ \) and \( 108^\circ \).

L M N O P Q 72° 108°

Prove, with a calculation and a reason, whether line \( LM \) is parallel to line \( NO \).

[3]
--- END OF PAPER ---
Mathwise Academy | Grade 8 Test 1 - Measurement (Pythagoras, Area, Perimeter, & Volume)
M

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Measurement: Pythagoras, Area, Perimeter, & Volume

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

  • Answer ALL the questions.
  • Read all questions carefully before you begin.
  • Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
  • You may use an approved scientific calculator.
  • Round off final answers to TWO decimal places where necessary, unless specified otherwise.
  • Use \( \pi = 3,14 \) or the \( \pi \) button on your calculator.
  • Write neatly and legibly.

QUESTION 1: THEOREM OF PYTHAGORAS [13]

1.1 Consider the right-angled triangle \( \triangle ABC \) below:

A B C x 24 cm 25 cm
  • 1.1.1 Calculate the length of side \( AB \) (indicated by \( x \)). (3)

1.2 Determine whether a triangle with sides \( 9\text{ cm} \), \( 12\text{ cm} \), and \( 16\text{ cm} \) is an acute-angled, obtuse-angled, or right-angled triangle. Show all your calculations to justify your answer.

[4]

1.3 Application Problem:
A kite is flying high in the air. The string of the kite is \( 50\text{ m} \) long. The child holding the string is standing on the ground, and the horizontal distance from the child to a point directly underneath the kite is \( 30\text{ m} \).

50 m (String) 30 m h Child Point on ground Kite
  • 1.3.1 Calculate the vertical height (\( h \)) of the kite above the ground. (3)
  • 1.3.2 If the child lets out another \( 15\text{ m} \) of string (total string length is now \( 65\text{ m} \)) while the vertical height (\( h \)) remains the same, calculate the new distance along the ground from the child to the point directly underneath the kite. (3)

QUESTION 2: AREA AND PERIMETER OF 2D SHAPES [20]

2.1 Convert the following units:

  • 2.1.1 \( 45\,000\text{ cm}^2 \) to \( \text{m}^2 \). (2)
  • 2.1.2 \( 3,5\text{ km} \) to \( \text{m} \). (1)

2.2 Consider the composite shape below. It consists of a rectangle and a semi-circle attached to one of its shorter sides.

12 cm 8 cm Diameter = 8 cm
  • 2.2.1 Calculate the perimeter of the entire shape. (4)
  • 2.2.2 Calculate the total area of the shape. (4)

2.3 In the diagram below, an isosceles triangle is drawn inside a square. The side length of the square is \( 10\text{ cm} \). The base of the triangle sits exactly on the bottom edge of the square, and its top vertex touches the midpoint of the top edge of the square.

10 cm 10 cm
  • 2.3.1 Calculate the area of the white triangle. (2)
  • 2.3.2 Hence, calculate the area of the shaded regions. (2)

2.4 A farmer has a circular plot of land with a diameter of \( 40\text{ m} \). He wants to put a fence around the exact edge of the plot. The fencing costs \( \text{R } 120 \) per metre.

Calculate the total cost to fence the circular plot.

[5]

QUESTION 3: SURFACE AREA AND VOLUME OF 3D SHAPES [17]

3.1 Convert the following units:

  • 3.1.1 \( 1\,500\,000\text{ cm}^3 \) to \( \text{m}^3 \). (2)
  • 3.1.2 \( 4,2\text{ litres} \) to \( \text{cm}^3 \). (2)

3.2 Consider the closed rectangular prism (cuboid) below:

25 cm 12 cm 10 cm
  • 3.2.1 Calculate the total surface area of the rectangular prism. (4)
  • 3.2.2 Calculate the volume of the rectangular prism. (2)
  • 3.2.3 Calculate the capacity of the prism in litres. (2)

3.3 A solid block of wood is shaped like a right-angled triangular prism. The dimensions of the triangular base are: base \( = 6\text{ cm} \), height \( = 8\text{ cm} \), and hypotenuse \( = 10\text{ cm} \). The length (depth) of the prism is \( 20\text{ cm} \).

8 cm 6 cm 10 cm 20 cm
  • 3.3.1 Calculate the volume of the triangular prism. (2)
  • 3.3.2 Calculate the total surface area of this closed triangular prism. (3)
--- END OF PAPER ---
Mathwise Academy | Grade 8 Test 2 - Measurement (Higher Order Applications)
M

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Measurement (Higher Order Applications)

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

  • Answer ALL the questions.
  • Read all questions carefully before you begin.
  • Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
  • You may use an approved scientific calculator.
  • Round off final answers to TWO decimal places where necessary, unless specified otherwise.
  • Use \( \pi = 3,14 \) or the \( \pi \) button on your calculator.
  • Write neatly and legibly.

QUESTION 1: THEOREM OF PYTHAGORAS [12]

1.1 The Sliding Ladder Problem:
A ladder that is \( 5\text{ m} \) long rests against a vertical wall. The foot of the ladder is initially \( 3\text{ m} \) away from the base of the wall. After a while, the ladder slips down the wall so that its foot is now \( 4\text{ m} \) away from the base of the wall.

5 m 5 m 3 m 4 m y Wall Floor

Calculate exactly how far down the wall the top of the ladder has slipped (indicated by distance \( y \)). Show all your working.

[4]

1.2 Consider the composite figure below consisting of two right-angled triangles, \( \triangle PQR \) and \( \triangle PRS \). Angle \( \hat{Q} \) is a right angle (\( 90^\circ \)) and angle \( P\hat{R}S \) is a right angle (\( 90^\circ \)).

P Q R S 15 cm 8 cm PR 10 cm x

Calculate the length of side \( PS \) (indicated by \( x \)).

[4]

1.3 A triangle has side lengths of \( 20\text{ mm} \), \( 21\text{ mm} \), and \( 29\text{ mm} \). Determine, showing all calculations, whether this is a right-angled triangle.

[4]

QUESTION 2: AREA AND PERIMETER OF 2D SHAPES [18]

2.1 Convert \( 5,2\text{ m}^2 \) to \( \text{cm}^2 \).

[2]

2.2 A rectangular sheet of metal has a length of \( 20\text{ cm} \) and a breadth of \( 14\text{ cm} \). A semi-circle is cut out of one of the shorter sides, as shown in the diagram below:

D C B A 20 cm 14 cm
  • 2.2.1 Calculate the perimeter of the remaining shape (the shaded region). (4)
  • 2.2.2 Calculate the area of the remaining shape (the shaded region). (4)

2.3 A circular flower bed has a radius of \( 6\text{ m} \). It is surrounded by a paved path that is \( 2\text{ m} \) wide all the way around.

6 m 2 m

Calculate the area of the paved path (the shaded ring).

[3]

2.4 Application Problem:
A rectangular garden measures \( 15\text{ m} \) by \( 10\text{ m} \). A concrete pathway of \( 1\text{ m} \) uniform width is built inside the garden, along all four borders.

Calculate the total area of this concrete pathway.

[5]

QUESTION 3: SURFACE AREA AND VOLUME OF 3D SHAPES [20]

3.1 A rectangular prism (cuboid) has a length of \( 12\text{ cm} \), a breadth of \( 8\text{ cm} \), and a height of \( 5\text{ cm} \).

Calculate the surface area of the prism if it is open on top (i.e. it has no lid).

[4]

3.2 Consider the right-angled triangular prism below. The dimensions of the triangular base are \( 6\text{ cm} \), \( 8\text{ cm} \), and \( 10\text{ cm} \). The length of the prism is \( 20\text{ cm} \).

8 cm 6 cm 10 cm 20 cm
  • 3.2.1 Calculate the volume of the triangular prism. (3)
  • 3.2.2 Calculate the total surface area of this closed triangular prism. (4)

3.3 Capacity Application Problem:
A large rectangular glass fish tank is \( 80\text{ cm} \) long, \( 40\text{ cm} \) wide, and \( 50\text{ cm} \) high.

  • 3.3.1 Calculate the total volume of the fish tank in \( \text{cm}^3 \). (2)
  • 3.3.2 Calculate the total capacity of the fish tank in litres. (2)
  • 3.3.3 If the tank is filled with water to a depth of \( 35\text{ cm} \), how many litres of water are currently in the tank? (3)

3.4 If the side length of a cube is doubled, by what factor does its volume increase?
(For example, does it double, triple, multiply by 4, 8, etc.?) Provide a brief explanation or calculation to prove your answer.

[2]
--- END OF PAPER ---
Mathwise Academy | Grade 8 Test 2 - Statistics and Transformation Geometry (Standard Level)
M

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Statistics and Transformation Geometry (Standard Level)

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

  • Answer ALL the questions.
  • Read all questions carefully before you begin.
  • Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
  • You may use an approved non-programmable scientific calculator.
  • Write neatly and legibly.

QUESTION 1: STATISTICS (DATA HANDLING) [25]

1.1 The following data represents the marks out of \( 20 \) achieved by a group of \( 9 \) learners in a short quiz:

12   ;   14   ;   11   ;   14   ;   18   ;   15   ;   12   ;   14   ;   16
  • 1.1.1 Arrange the data in ascending order. (1)
  • 1.1.2 Write down the mode of the dataset. (1)
  • 1.1.3 Determine the median mark. (2)
  • 1.1.4 Calculate the mean (average) mark for this group of learners. (3)
  • 1.1.5 Calculate the range of the marks. (1)

1.2 The stem-and-leaf plot below shows the final Mathematics test scores (out of \( 50 \)) achieved by a group of \( 15 \) learners:

Stem Leaf
1 2   5   8
2 1   4   4   4   7
3 0   5   9
4 2   2   8   9
  • 1.2.1 Write down the lowest score achieved in the test. (1)
  • 1.2.2 Write down the mode of the data. (1)
  • 1.2.3 Determine the median score. (2)
  • 1.2.4 How many learners scored \( 40 \) marks or more? (1)

1.3 The pie chart below represents the monthly household budget of the Khumalo family. The total monthly budget is \( \text{R } 12\,000 \). The angles for the different sectors are shown on the chart:

Rent 120° Food 90° Transport 60° 90° Savings
  • 1.3.1 What fraction of the budget is spent on Rent? Write your answer in its simplest form. (2)
  • 1.3.2 Calculate the exact amount of money the Khumalo family spends on Food. (2)
  • 1.3.3 Calculate the amount of money put into Savings. (1)

1.4 A Grade 8 class was surveyed to find out what their favorite fruits were. The results are shown in the bar graph below:

0 2 4 6 8 10 12 Number of Learners Apple Banana Orange Grapes Fruit
  • 1.4.1 Which fruit is the most popular in the class? (1)
  • 1.4.2 How many learners participated in this survey in total? (2)
  • 1.4.3 What percentage of the class chose Banana as their favorite fruit? (2)
  • 1.4.4 Determine the ratio of learners who chose Orange to learners who chose Apple, in its simplest form. (2)

QUESTION 2: TRANSFORMATION GEOMETRY [25]

2.1 A point \( P \) has coordinates \( (2 ; 5) \).

  • 2.1.1 Point \( P \) is translated \( 4 \) units to the left and \( 1 \) unit down to become \( P' \). Write down the coordinates of \( P' \). (2)

2.2 A point \( Q \) has coordinates \( (-3 ; 4) \).

  • 2.2.1 Point \( Q \) is reflected across the \( x \)-axis to become \( Q' \). Write down the coordinates of \( Q' \). (2)

2.3 Study the Cartesian plane below. \( \triangle DEF \) has been transformed to create its image, \( \triangle D'E'F' \):

x y -4 -2 2 4 -2 -4 2 4 D E F D' E' F'

Describe fully the translation that maps \( \triangle DEF \) onto \( \triangle D'E'F' \).
(State how many units it moved left/right and how many units up/down).

[3]

2.4 The general rule for rotating a point \( 90^\circ \) clockwise about the origin is \( (x ; y) \rightarrow (y ; -x) \). A line segment \( AB \) has endpoints at \( A(1 ; 3) \) and \( B(4 ; 5) \).

Apply this rule to determine the coordinates of the image endpoints \( A' \) and \( B' \).

[4]

2.5 Enlargement (Scale Factor):
A rectangle has a length of \( 5\text{ cm} \) and a width of \( 3\text{ cm} \). The rectangle is enlarged by a scale factor of \( k = 2 \).

  • 2.5.1 What are the dimensions (new length and new width) of the enlarged rectangle? (2)
  • 2.5.2 Calculate the perimeter of the original rectangle AND the perimeter of the enlarged rectangle. (4)
  • 2.5.3 Calculate the area of the original rectangle AND the area of the enlarged rectangle. (4)

2.6 Study the diagram below showing \( \triangle KLM \) and its image \( \triangle K'L'M' \):

x y -3 -2 -1 1 2 3 2 4 K L M L' K' M'

What single geometric transformation maps \( \triangle KLM \) onto \( \triangle K'L'M' \)?

[2]

2.7 How many lines of symmetry does a regular pentagon (a shape with \( 5 \) equal sides) have?

[2]
--- END OF PAPER ---
Mathwise Academy | Grade 8 Test 2 - Statistics and Transformation Geometry (Standard Level)
M

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Statistics and Transformation Geometry (Standard Level)

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

  • Answer ALL the questions.
  • Read all questions carefully before you begin.
  • Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
  • You may use an approved non-programmable scientific calculator.
  • Write neatly and legibly.

QUESTION 1: STATISTICS (DATA HANDLING) [25]

1.1 The following data represents the marks out of \( 20 \) achieved by a group of \( 9 \) learners in a short quiz:

12   ;   14   ;   11   ;   14   ;   18   ;   15   ;   12   ;   14   ;   16
  • 1.1.1 Arrange the data in ascending order. (1)
  • 1.1.2 Write down the mode of the dataset. (1)
  • 1.1.3 Determine the median mark. (2)
  • 1.1.4 Calculate the mean (average) mark for this group of learners. (3)
  • 1.1.5 Calculate the range of the marks. (1)

1.2 The stem-and-leaf plot below shows the final Mathematics test scores (out of \( 50 \)) achieved by a group of \( 15 \) learners:

Stem Leaf
1 2   5   8
2 1   4   4   4   7
3 0   5   9
4 2   2   8   9
  • 1.2.1 Write down the lowest score achieved in the test. (1)
  • 1.2.2 Write down the mode of the data. (1)
  • 1.2.3 Determine the median score. (2)
  • 1.2.4 How many learners scored \( 40 \) marks or more? (1)

1.3 The pie chart below represents the monthly household budget of the Khumalo family. The total monthly budget is \( \text{R } 12\,000 \). The angles for the different sectors are shown on the chart:

Rent 120° Food 90° Transport 60° 90° Savings
  • 1.3.1 What fraction of the budget is spent on Rent? Write your answer in its simplest form. (2)
  • 1.3.2 Calculate the exact amount of money the Khumalo family spends on Food. (2)
  • 1.3.3 Calculate the amount of money put into Savings. (1)

1.4 A Grade 8 class was surveyed to find out what their favorite fruits were. The results are shown in the bar graph below:

0 2 4 6 8 10 12 Number of Learners Apple Banana Orange Grapes Fruit
  • 1.4.1 Which fruit is the most popular in the class? (1)
  • 1.4.2 How many learners participated in this survey in total? (2)
  • 1.4.3 What percentage of the class chose Banana as their favorite fruit? (2)
  • 1.4.4 Determine the ratio of learners who chose Orange to learners who chose Apple, in its simplest form. (2)

QUESTION 2: TRANSFORMATION GEOMETRY [25]

2.1 A point \( P \) has coordinates \( (2 ; 5) \).

  • 2.1.1 Point \( P \) is translated \( 4 \) units to the left and \( 1 \) unit down to become \( P' \). Write down the coordinates of \( P' \). (2)

2.2 A point \( Q \) has coordinates \( (-3 ; 4) \).

  • 2.2.1 Point \( Q \) is reflected across the \( x \)-axis to become \( Q' \). Write down the coordinates of \( Q' \). (2)

2.3 Study the Cartesian plane below. \( \triangle DEF \) has been transformed to create its image, \( \triangle D'E'F' \):

x y -4 -2 2 4 -2 -4 2 4 D E F D' E' F'

Describe fully the translation that maps \( \triangle DEF \) onto \( \triangle D'E'F' \).
(State how many units it moved left/right and how many units up/down).

[3]

2.4 The general rule for rotating a point \( 90^\circ \) clockwise about the origin is \( (x ; y) \rightarrow (y ; -x) \). A line segment \( AB \) has endpoints at \( A(1 ; 3) \) and \( B(4 ; 5) \).

Apply this rule to determine the coordinates of the image endpoints \( A' \) and \( B' \).

[4]

2.5 Enlargement (Scale Factor):
A rectangle has a length of \( 5\text{ cm} \) and a width of \( 3\text{ cm} \). The rectangle is enlarged by a scale factor of \( k = 2 \).

  • 2.5.1 What are the dimensions (new length and new width) of the enlarged rectangle? (2)
  • 2.5.2 Calculate the perimeter of the original rectangle AND the perimeter of the enlarged rectangle. (4)
  • 2.5.3 Calculate the area of the original rectangle AND the area of the enlarged rectangle. (4)

2.6 Study the diagram below showing \( \triangle KLM \) and its image \( \triangle K'L'M' \):

x y -3 -2 -1 1 2 3 2 4 K L M L' K' M'

What single geometric transformation maps \( \triangle KLM \) onto \( \triangle K'L'M' \)?

[2]

2.7 How many lines of symmetry does a regular pentagon (a shape with \( 5 \) equal sides) have?

[2]
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