How to Use This Page
1. Attempt each exercise problem on your own first. 2. Compare your method and answer with the step-by-step solution provided below. 3. If you made an error, identify the step where you went wrong and redo the problem. 4. Revisit the Key Concepts section for quick formula revision before exams.
Key Concepts (Quick Recap)
- Translation: A slide that moves every point of a shape by the same distance in the same direction. Described as "x right/left, y up/down."
- Reflection: A flip over a line (called the line of reflection or mirror line). Every point and its image are equidistant from the mirror line.
- Rotation: A turn about a fixed point (centre of rotation) through a given angle (e.g., 90 degrees, 180 degrees, 270 degrees).
- Circle: A set of all points equidistant from a fixed point called the centre. Radius (r) is the distance from centre to any point on the circle. Diameter (d) = 2r.
- Concentric Circles: Circles that share the same centre but have different radii.
- Width of Circular Ring: When two concentric circles have outer radius R and inner radius r, the width of the ring = R - r.
- Corresponding Parts: Under any transformation, vertices map to vertices (e.g., L maps to L'), and sides map to corresponding sides.
Exercise Solutions
Exercise 4.1 — Symmetry and Transformations
Question 1
The question is image-based (identifying symmetries in figures).
Refer to the textbook figures and apply the concept of lines of symmetry.
Question 2: How is the pre-image translated to the image?
Answer
Describe the translation for each part.
(i) Translation: 3 right, 4 up
Every point of the pre-image shifts 3 units to the right and 4 units upward.
(ii) Translation: 3 left, 3 up
Every point of the pre-image shifts 3 units to the left and 3 units upward.
(iii) Translation: 4 left, 4 down
Every point of the pre-image shifts 4 units to the left and 4 units downward.
(iv) Translation: 2 left, 2 down
Every point of the pre-image shifts 2 units to the left and 2 units downward.
Question 3: Find the image of the given triangle with given translation
Answer
This is an image-based construction question. To find the image of a triangle under a translation:
Step 1: Identify the translation (e.g., "3 right, 2 up").
Step 2: Take each vertex of the triangle.
Step 3: Shift each vertex by the given translation.
Step 4: Join the new vertices to form the image triangle.
Question 4: Reflect the shape with given line of reflection
Answer
This is an image-based construction question. Steps to reflect a shape:
Step 1: Identify the mirror line (line of reflection).
Step 2: From each vertex, draw a perpendicular to the mirror line.
Step 3: Extend each perpendicular beyond the mirror line by the same distance.
Step 4: Mark the new points and join them to form the reflected image.
Question 5: Reflect the shape in each picture with given line of reflection
Answer
Image-based. Apply the reflection method described in Question 4 for each figure.
Question 6: Rotate the preimages about the red point
Answer
Image-based. Steps to rotate a figure:
Step 1: Identify the centre of rotation (the red point).
Step 2: Identify the angle of rotation (e.g., 90 degrees clockwise).
Step 3: From the centre, draw lines to each vertex.
Step 4: Rotate each line by the given angle.
Step 5: Mark the new vertex positions and join them.
Question 7: Identify the Transformation
AnswerReflection
The figure shows a flip over a line; each point and its image are on opposite sides of the mirror line.
Question 8: Identify the Transformation
AnswerRotation
The figure shows a turn about a point; the shape has been rotated rather than slid or flipped.
Question 9: Identify the Transformation
AnswerTranslation
The figure shows a slide; every point has moved the same distance in the same direction without turning or flipping.
Question 10: A pool of fish translates from point F to point D
Answer
(a) Describe the translation of the pool of fish.
AnswerTranslation = 7 right, 2 down
Every point of the fish shape shifts 7 units to the right and 2 units downward.
(b) Can the fishing boat make the same translation? Explain.
AnswerNo. If the fishing boat translates 7 right and 2 down, it would land on the island, which is not a valid position.
(c) Describe a translation the fishing boat could make to get to point D.
AnswerTranslation = 5 right, 3 down
The boat moves 5 units right and 3 units down to reach point D.
Question 11: Name the transformation that maps footprint A onto each footprint
Answer
(i) Footprint B → Translation (the shape slides without turning or flipping)
(ii) Footprint C → Reflection about a horizontal line (the shape flips over a horizontal mirror line)
(iii) Footprint D → Reflection about a vertical line (the shape flips over a vertical mirror line)
(iv) Footprint E → Rotation about the heel (the shape turns around the heel point)
Question 12: Blue figure is the image of the pink figure
Answer
(i) Choose an angle or vertex from the pre-image and name its image.
Angles: angle L maps to angle L'
angle M maps to angle M'
angle N maps to angle N'
angle O maps to angle O'
Vertices: L maps to L'
M maps to M'
N maps to N'
O maps to O'
(ii) List all pairs of corresponding sides.
Side LM maps to side L'M'
Side MN maps to side M'N'
Side NO maps to side N'O'
Side OL maps to side O'L'
Question 13: Coordinate rule for translation
Answer
The green figure is a translation image of the pink figure.
AnswerThe translation rule is 3 right, 1 down
Every point (x, y) of the pink figure maps to (x + 3, y - 1) in the green figure.
Objective Type Questions
Q1. A ______ is a turn about a point.
Answer(ii) Rotation
A rotation is defined as a turn about a fixed point.
Q2. A ______ is a flip over a line.
Answer(iii) Reflection
A reflection is defined as a flip over a line (mirror line).
Q3. A ______ is a slide; move without turning or flipping the shape.
Answer(i) Translation
A translation slides every point by the same distance in the same direction.
Q4. The transformation used in the picture is:
Answer(ii) Rotation
Q5. The transformation used in the picture is:
Answer(i) Translation
Q6. You must rotate the puzzle piece 270 degrees clockwise about point P to fit it into a puzzle. Which piece fits?
(Image-based question. The correct piece is the one that, when rotated 270 degrees clockwise, aligns with the gap in the puzzle.)
Exercise 4.2 — Circles and Concentric Circles
Question 1: Draw circles for the following measurements of radius (r) / diameter (d)
Answer
(i) r = 4 cm
Step 1: Mark a point O on the paper as the centre.
Step 2: Set the compass distance to the radius = 4 cm.
Step 3: Place the pointed end of the compass on point O.
Step 4: Hold the compass firmly at O and slowly rotate it to draw the circle.
(ii) d = 12 cm
Given: d = 12 cm
Step 1: Find the radius. r = d / 2 = 12 / 2 = 6 cm
Step 2: Mark a point O on the paper as the centre.
Step 3: Set the compass distance to the radius = 6 cm.
Step 4: Place the pointed end of the compass on point O.
Step 5: Hold the compass firmly at O and slowly rotate it to draw the circle.
(iii) r = 3.5 cm
Step 1: Mark a point O on the paper as the centre.
Step 2: Set the compass distance to the radius = 3.5 cm.
Step 3: Place the pointed end of the compass on point O.
Step 4: Hold the compass firmly at O and slowly rotate it to draw the circle.
(iv) r = 6.5 cm
Step 1: Mark a point O on the paper as the centre.
Step 2: Set the compass distance to the radius = 6.5 cm.
Step 3: Place the pointed end of the compass on point O.
Step 4: Hold the compass firmly at O and slowly rotate it to draw the circle.
(v) d = 6 cm
Given: d = 6 cm
Step 1: Find the radius. r = d / 2 = 6 / 2 = 3 cm
Step 2: Mark a point O on the paper as the centre.
Step 3: Set the compass distance to the radius = 3 cm.
Step 4: Place the pointed end of the compass on point O.
Step 5: Hold the compass firmly at O and slowly rotate it to draw the circle.
Question 2: Draw concentric circles and find the width of each circular ring
Answer
(i) r = 3 cm and r = 5 cm
Given: Inner radius r = 3 cm, Outer radius R = 5 cm
Step 1: Mark a point O on the paper as the common centre.
Step 2: With O as centre, draw a circle of radius OA = 3 cm.
Step 3: With O as centre, draw a circle of radius OB = 5 cm.
Step 4: Width of the circular ring = R - r = 5 - 3 = 2 cm
Answer: Width = 2 cm
(ii) r = 3.5 cm and r = 6.5 cm
Given: Inner radius r = 3.5 cm, Outer radius R = 6.5 cm
Step 1: Mark a point O on the paper as the common centre.
Step 2: With O as centre, draw a circle of radius 3.5 cm.
Step 3: With O as centre, draw a circle of radius 6.5 cm.
Step 4: Width of the circular ring = R - r = 6.5 - 3.5 = 3 cm
Answer: Width = 3 cm
(iii) d = 6.4 cm and d = 11.6 cm
Given: Inner diameter = 6.4 cm, Outer diameter = 11.6 cm
Step 1: Convert diameters to radii.
Inner radius r = 6.4 / 2 = 3.2 cm
Outer radius R = 11.6 / 2 = 5.8 cm
Step 2: Mark a point O on the paper as the common centre.
Step 3: With O as centre, draw a circle of radius 3.2 cm.
Step 4: With O as centre, draw a circle of radius 5.8 cm.
Step 5: Width of the circular ring = R - r = 5.8 - 3.2 = 2.6 cm
Answer: Width = 2.6 cm
(iv) r = 5 cm and r = 7.5 cm
Given: Inner radius r = 5 cm, Outer radius R = 7.5 cm
Step 1: Mark a point O on the paper as the common centre.
Step 2: With O as centre, draw a circle of radius 5 cm.
Step 3: With O as centre, draw a circle of radius 7.5 cm.
Step 4: Width of the circular ring = R - r = 7.5 - 5 = 2.5 cm
Answer: Width = 2.5 cm
(v) d = 6.2 cm and r = 6.2 cm
Given: Inner diameter = 6.2 cm, Outer radius = 6.2 cm
Step 1: Convert the diameter to radius.
Inner radius r = 6.2 / 2 = 3.1 cm
Outer radius R = 6.2 cm
Step 2: Mark a point O on the paper as the common centre.
Step 3: With O as centre, draw a circle of radius 3.1 cm.
Step 4: With O as centre, draw a circle of radius 6.2 cm.
Step 5: Width of the circular ring = R - r = 6.2 - 3.1 = 3.1 cm
Answer: Width = 3.1 cm
(vi) r = 7.1 cm and d = 12 cm
Given: One radius = 7.1 cm, Diameter = 12 cm
Step 1: Convert the diameter to radius.
r from diameter = 12 / 2 = 6 cm
Step 2: The two radii are 6 cm and 7.1 cm.
Inner radius r = 6 cm, Outer radius R = 7.1 cm
Step 3: Mark a point O on the paper as the common centre.
Step 4: With O as centre, draw a circle of radius 6 cm.
Step 5: With O as centre, draw a circle of radius 7.1 cm.
Step 6: Width of the circular ring = R - r = 7.1 - 6 = 1.1 cm
Answer: Width = 1.1 cm
Exercise 4.3 — Miscellaneous and Challenge Problems
Miscellaneous Practice Problems
Q1: The bishop in a chess board can move diagonally along dark squares. Describe the translations after two moves shown in the figure.
First move: 2 right, 2 down
Second move: 5 left, 5 down
The bishop moves diagonally, so each move has equal horizontal and vertical components.
Q2: Write a possible translation for each chess piece for a single move.
Pawn: 1 up (or 2 up for its first move)
Rook: 1 to 8 squares in any one direction (up, down, left, or right)
Knight: 2 right & 1 up; 2 left & 1 up; 1 right & 2 up; 1 left & 2 up
(and similar L-shaped moves in other directions)
Bishop: 1 to 5 right & 1 to 5 up (or left & up, right & down, left & down)
(diagonal moves of varying lengths)
Queen: 1 to 8 up (or any direction, including diagonals)
King: 1 right, 1 left, 1 up, 1 down, or 1 diagonally
Q3: Each bar of the category is a boy-girl-boy unit.
(i) Which categories show translation within the bar?
AnswerEssay Writing shows translation within the bar.
(ii) Which categories show reflection within the bar?
AnswerEssay Writing and Mono Acting show reflection within the bar.
Q4: A floor design has small red equilateral triangles of side 30 cm. Describe translations.
(i) Yellow line: 120 cm right, 210 cm down
(ii) Black line: 270 cm left, 330 cm up
(iii) Blue line: 150 cm right
Q5: Describe the transformation in each pair of letter figures.
(i) Rotation
(ii) Reflection
(iii) Translation
(iv) Reflection
(v) Rotation
(vi) Reflection
(vii) Rotation
(viii) Translation
Challenge Problems
Q1: In chess, a knight moves in an L-shaped pattern. Write translations to move the knight from g8 to g5 in at most two moves.
Given: Knight at g8, target g5.
The knight needs to move from column g to column g (same column) and from row 8 to row 5 (3 rows down).
Move 1: 2 left, 1 down (from g8 to e7)
Move 2: 1 left, 2 down (from e7 to d5) — wait, this doesn't reach g5.
Alternative:
Move 1: 2 left, 1 down (from g8 to e7) — no, let's reconsider.
Better approach:
Move 1: 2 squares in one direction and 1 in perpendicular.
From g8: go 2 left and 1 down = e7
From e7: go 2 right and 1 down = g5 (reached!)
Answer: First move: 2 left, 1 down (g8 to e7). Second move: 2 right, 1 down (e7 to g5).
Q2: The pink shape is congruent to the blue shape. Describe a sequence of transformations.
Method (i):
Step 1: Translate 3 left and 5 up
Step 2: Rotate 90 degrees counter-clockwise about the green point
Step 3: Translate 5 left and 2 down
Method (ii):
Step 1: Translate 2 left
Step 2: Rotate 90 degrees counter-clockwise about the green point
Step 3: Translate 2 left and 2 down
Q3: Image-based question. Refer to the textbook for the complete figure and solution.
Q4: Draw concentric circles with inner circle radius 4.5 cm and circular ring width 2.5 cm.
Given: Inner radius r = 4.5 cm, Ring width = 2.5 cm
Step 1: Find the outer radius.
Outer radius R = r + ring width = 4.5 + 2.5 = 7 cm
Step 2: Mark a point O on the paper as the common centre.
Step 3: With O as centre, draw a circle of radius 4.5 cm (inner circle).
Step 4: With O as centre, draw a circle of radius 7 cm (outer circle).
Step 5: Verify: Width of ring = 7 - 4.5 = 2.5 cm (correct).
Answer: Outer radius = 7 cm
Q5: Draw concentric circles with outer circle radius 5.3 cm and circular ring width 1.8 cm.
Given: Outer radius R = 5.3 cm, Ring width = 1.8 cm
Step 1: Find the inner radius.
Inner radius r = R - ring width = 5.3 - 1.8 = 3.5 cm
Step 2: Mark a point O on the paper as the common centre.
Step 3: With O as centre, draw a circle of radius 3.5 cm (inner circle).
Step 4: With O as centre, draw a circle of radius 5.3 cm (outer circle).
Step 5: Verify: Width of ring = 5.3 - 3.5 = 1.8 cm (correct).
Answer: Inner radius = 3.5 cm
Common Mistakes
1. Confusing translation and rotation: A translation is a slide (every point moves the same distance in the same direction). A rotation is a turn about a fixed point. If the shape turns, it is not a translation.
2. Wrong direction in translation descriptions: Always specify both direction and magnitude. "3 right, 4 up" is correct; just writing "3, 4" is incomplete. Use arrows or words like left/right and up/down.
3. Forgetting to convert diameter to radius: When a diameter is given, always compute radius first: r = d / 2. Forgetting this step gives a circle twice the required size.
4. Calculating ring width incorrectly: The width of a circular ring is the difference between the outer radius and the inner radius (R - r), not the difference of diameters.
5. Misidentifying the mirror line in reflections: The mirror line is the perpendicular bisector of the segment joining any point and its image. Every point on the original figure and its reflected image are equidistant from the mirror line.
6. Describing knight moves incorrectly: A knight always moves in an L-shape: 2 squares in one direction and 1 square perpendicular (or 1 square in one direction and 2 squares perpendicular). It cannot move in a straight line.
7. Mixing up clockwise and counter-clockwise rotation: A positive rotation (counter-clockwise) goes in the opposite direction to clock hands. Always specify the direction when describing rotations.
Exam Checklist
- [ ] Can you identify whether a transformation is a translation, reflection, or rotation?
- [ ] Can you describe a translation using "x right/left, y up/down"?
- [ ] Can you find the image of a point or shape after a given translation?
- [ ] Can you draw the reflection of a shape across a given mirror line?
- [ ] Can you rotate a shape about a given point by a specified angle?
- [ ] Can you draw a circle with a given radius or diameter using a compass?
- [ ] Can you draw concentric circles and calculate the width of a circular ring?
- [ ] Do you remember: radius = diameter / 2 and diameter = 2 x radius?
- [ ] Can you describe the movement of chess pieces in terms of translations?
- [ ] Can you combine two or more transformations (translation followed by rotation, etc.)?
