Class 8 Mathematics - Ch 8: Geometric Transformations Mastery Guide (FBISE)
Teacher & Parent Instructional Roadmap
Unit Focus: Under the Pakistan Single National Curriculum (SNC 2022) and Federal Board (FBISE) standards for Grade 8 Mathematics, Unit 8: Transformations (تحویلات / تبدیلیاں) bridges synthetic 2D Euclidean geometry and analytic coordinate algebra. Students learn to classify, calculate, and graph translations, reflections across axes, coordinate plane rotations about the origin, and proportional dilations.
- Define and differentiate the 4 fundamental transformations: Translation (slide), Reflection (flip), Rotation (turn), and Dilation (resizing).
- Apply coordinate reflection rules: across $x$-axis $(x, y) \to (x, -y)$ and $y$-axis $(x, y) \to (-x, y)$.
- Identify vertical and horizontal line symmetry in capital alphabets.
- Execute $90^\circ$ clockwise $(y, -x)$, $90^\circ$ anti-clockwise $(-y, x)$, and $180^\circ$ $(-x, -y)$ rotations about the origin $(0,0)$.
- Scale 2D figures using dilation rule $(x, y) \to (kx, ky)$ for enlargement ($k > 1$) and reduction ($0 < k < 1$).
- Distinguish Rigid Transformations / Isometries (congruent $\cong$) from Non-Rigid Transformations (similar $\sim$).
- Confusing Reflection Axes: In reflection over the $x$-axis ($y=0$), the $x$-coordinate stays constant and $y$ flips sign: $(x, y) \to (x, -y)$. Over $y$-axis: $(x, y) \to (-x, y)$.
- Mixing CW vs. CCW Rotations: $90^\circ$ Clockwise is $(y, -x)$, whereas $90^\circ$ Counter-Clockwise (Anti-Clockwise) is $(-y, x)$.
- Dilation Area Misconception: If side lengths enlarge by $k=2$, area scales by $k^2 = 4$ times (not 2 times).
- Non-Origin Centers: When dilating from a polygon vertex (e.g. $A$), that vertex remains invariant ($A' = A$) while remaining vertices scale outwards from it.
2. Apply Coordinate Mapping Formula: Apply the exact formula $(x, y) \to (x', y')$ to each vertex of the pre-image step-by-step.
3. Verify Invariant Properties: Confirm side length and area conservation for isometries (congruence $\cong$) or shape and angle conservation for dilations (similarity $\sim$).
🌍 Real-World Connections & Kid-Friendly Tips
🌟 Comprehensive Theory & Coordinate Rules
1. Translation (Vector Slide):
Moves all points of a shape by a fixed horizontal shift $h$ and vertical shift $k$:
$$(x, y) \longrightarrow (x + h, y + k)$$
Right: $+h$, Left: $-h$, Up: $+k$, Down: $-k$.
2. Reflection (Mirror Inversion):
Flips a shape across a line of reflection so that every point and its image are equidistant from the mirror line:
$$\text{Reflection across } x\text{-axis: } (x, y) \longrightarrow (x, -y)$$
$$\text{Reflection across } y\text{-axis: } (x, y) \longrightarrow (-x, y)$$
Alphabet Line Symmetry:
- Vertical Line Symmetry: A, H, I, M, O, T, U, V, W, X, Y
- Horizontal Line Symmetry: B, C, D, E, H, I, K, O, X
- Both Vertical & Horizontal: H, I, O, X
- No Line Symmetry: F, G, J, L, N, P, Q, R, S, Z
3. Rotation (Turn about Origin):
Turns a shape by angle $\theta$ around the centre of rotation $(0,0)$:
$$\text{Rotation } 90^\circ \text{ Clockwise (CW): } (x, y) \longrightarrow (y, -x)$$
$$\text{Rotation } 90^\circ \text{ Counter-Clockwise (CCW): } (x, y) \longrightarrow (-y, x)$$
$$\text{Rotation } 180^\circ \text{ (Half-turn): } (x, y) \longrightarrow (-x, -y)$$
$$\text{Rotation } 270^\circ \text{ Clockwise} \equiv 90^\circ \text{ Counter-Clockwise: } (x, y) \longrightarrow (-y, x)$$
4. Dilation (Proportional Resizing):
Resizes a shape by constant scale factor $k = \frac{\text{Image Length}}{\text{Pre-image Length}}$:
$$(x, y) \longrightarrow (kx, ky) \quad (k \neq 0)$$
$k > 1 \implies$ Enlargement | $0 < k < 1 \implies$ Reduction | $k = 1 \implies$ Congruent (no size change).
Exercise 8.1 • Reflections & Line Symmetry
Page 138Q1. On a coordinate plane, graph trapezium DEFG with vertices $D(1, 4)$, $E(6, 3)$, $F(4, 0)$, and $G(1, 0)$. Draw the image DEFG after a reflection across y-axis and x-axis.
Step 1: Original Coordinates:
Given pre-image vertices: $D(1, 4), E(6, 3), F(4, 0), G(1, 0)$.
Step 2: Reflection across y-axis $(x, y) \to (-x, y)$:
$$D(1, 4) \to D_1'(-1, 4), \quad E(6, 3) \to E_1'(-6, 3), \quad F(4, 0) \to F_1'(-4, 0), \quad G(1, 0) \to G_1'(-1, 0)$$
Step 3: Reflection across x-axis $(x, y) \to (x, -y)$:
$$D(1, 4) \to D_2'(1, -4), \quad E(6, 3) \to E_2'(6, -3), \quad F(4, 0) \to F_2'(4, 0), \quad G(1, 0) \to G_2'(1, 0)$$
Final Answer:
y-axis reflection image: $D'(-1, 4), E'(-6, 3), F'(-4, 0), G'(-1, 0)$
x-axis reflection image: $D''(1, -4), E''(6, -3), F''(4, 0), G''(1, 0)$
Q2. The letter A is a reflection of itself across the vertical line shown. Which other letters have this property?
Step 1: Understand Vertical Line Symmetry:
A letter has vertical symmetry if a central vertical line divides it into identical left and right mirror halves.
Step 2: Check all 26 alphabets:
Testing all capital letters reveals that A, H, I, M, O, T, U, V, W, X, and Y possess vertical line symmetry.
Final Answer: The other capital letters having vertical line symmetry are: H, I, M, O, T, U, V, W, X, Y.
Q3. Determine which of the capital letters have: (i) Horizontal line symmetry? (ii) Vertical line symmetry? (iii) No symmetry at all?
Step 1 (Horizontal symmetry): A horizontal mirror line creates identical top/bottom halves: B, C, D, E, H, I, K, O, X.
Step 2 (Vertical symmetry): A vertical mirror line creates identical left/right halves: A, H, I, M, O, T, U, V, W, X, Y.
Step 3 (No symmetry): Letters with neither mirror line: F, G, J, L, N, P, Q, R, S, Z.
Final Answer:
(i) Horizontal: B, C, D, E, H, I, K, O, X
(ii) Vertical: A, H, I, M, O, T, U, V, W, X, Y
(iii) No symmetry: F, G, J, L, N, P, Q, R, S, Z
Q4. On graph paper, draw a $\Delta ABC$ of your choice and its image $\Delta A'B'C'$ when $\Delta ABC$ is reflected across: (i) x-axis (ii) y-axis.
Step 1: Choose benchmark vertices: Let $A(1, 2), B(4, 2), C(2, 5)$.
Step 2 (Across x-axis): $(x, y) \to (x, -y) \implies A'(1, -2), B'(4, -2), C'(2, -5)$.
Step 3 (Across y-axis): $(x, y) \to (-x, y) \implies A''(-1, 2), B''(-4, 2), C''(-2, 5)$.
Final Answer: (i) $A'(1, -2), B'(4, -2), C'(2, -5)$ | (ii) $A''(-1, 2), B''(-4, 2), C''(-2, 5)$
Q5. On graph paper, draw a $\Delta ABC$ with vertices $A(-2, 0), B(2, 0), C(0, 3)$ and its image $\Delta A'B'C'$ when $\Delta ABC$ is reflected across x-axis.
Step 1: Apply reflection rule across x-axis: $(x, y) \to (x, -y)$.
Step 2: Transform vertices:
$A(-2, 0) \to A'(-2, 0)$ (Invariant, lies on $x$-axis)
$B(2, 0) \to B'(2, 0)$ (Invariant, lies on $x$-axis)
$C(0, 3) \to C'(0, -3)$
Final Answer: $A'(-2, 0), B'(2, 0), C'(0, -3)$
Q6. On graph paper, draw a $\Delta ABC$ with vertices $A(0, -2), B(0, 2), C(3, 0)$ and its image $\Delta A'B'C'$ when $\Delta ABC$ is reflected across y-axis.
Step 1: Apply reflection rule across y-axis: $(x, y) \to (-x, y)$.
Step 2: Transform vertices:
$A(0, -2) \to A'(0, -2)$ (Invariant, lies on $y$-axis)
$B(0, 2) \to B'(0, 2)$ (Invariant, lies on $y$-axis)
$C(3, 0) \to C'(-3, 0)$
Final Answer: $A'(0, -2), B'(0, 2), C'(-3, 0)$
Q7. Find the image of the point under a reflection across the given axis: (i) $(4, 1)$ across x-axis (ii) $(-2, 5)$ across y-axis (iii) $(7, -1)$ across y-axis (iv) $(10, 5)$ across y-axis.
(i) $(x, y) \to (x, -y) \implies (4, 1) \to (4, -1)$
(ii) $(x, y) \to (-x, y) \implies (-2, 5) \to (2, 5)$
(iii) $(x, y) \to (-x, y) \implies (7, -1) \to (-7, -1)$
(iv) $(x, y) \to (-x, y) \implies (10, 5) \to (-10, 5)$
Final Answer: (i) $(4, -1)$ | (ii) $(2, 5)$ | (iii) $(-7, -1)$ | (iv) $(-10, 5)$
Q8. Draw the reflection of: (i) $\Delta ABC$ with vertices $(2, 3), (3, 1), (5, 2)$ across x-axis. (ii) Square ABCD with vertices $(-2, 1), (-4, 1), (-4, 3), (-2, 3)$ across y-axis.
(i) Triangle across x-axis: $(x, y) \to (x, -y) \implies A'(2, -3), B'(3, -1), C'(5, -2)$.
(ii) Square across y-axis: $(x, y) \to (-x, y) \implies A'(2, 1), B'(4, 1), C'(4, 3), D'(2, 3)$.
Final Answer:
(i) Triangle Image: $A'(2, -3), B'(3, -1), C'(5, -2)$
(ii) Square Image: $A'(2, 1), B'(4, 1), C'(4, 3), D'(2, 3)$
Exercise 8.2 • Rotations, Dilations & Scale Factors
Page 143Q1. Triangle ABC has vertices $A(0, 0), B(3, 2), C(2, 3)$. Draw the image $A'B'C'$ after a clockwise turn of $90^\circ$ about the origin.
Step 1: Rule for $90^\circ$ CW turn about origin: $(x, y) \to (y, -x)$.
Step 2: Transform vertices:
$A(0, 0) \to A'(0, 0)$
$B(3, 2) \to B'(2, -3)$
$C(2, 3) \to C'(3, -2)$
Final Answer: $A'(0, 0), B'(2, -3), C'(3, -2)$
Q2. Repeat Q1 with origin as centre for: (i) Anti-clockwise rotation of $90^\circ$ (ii) Clockwise rotation of $180^\circ$.
(i) $90^\circ$ Anti-clockwise rule $(x, y) \to (-y, x)$:
$A(0, 0) \to A'(0, 0), \quad B(3, 2) \to B'(-2, 3), \quad C(2, 3) \to C'(-3, 2)$.
(ii) $180^\circ$ Rotation rule $(x, y) \to (-x, -y)$:
$A(0, 0) \to A''(0, 0), \quad B(3, 2) \to B''(-3, -2), \quad C(2, 3) \to C''(-2, -3)$.
Final Answer:
(i) $90^\circ$ CCW: $A'(0, 0), B'(-2, 3), C'(-3, 2)$
(ii) $180^\circ$ Turn: $A''(0, 0), B''(-3, -2), C''(-2, -3)$
Q3. Use the rule $(x, y) \to (-y, x)$ for a counter-clockwise rotation of $90^\circ$ of the triangle with vertices $A(0, 0), B(0, 3), C(3, 3)$. Draw the image of $\Delta ABC$.
Transform vertices:
$A(0, 0) \to A'(0, 0)$
$B(0, 3) \to B'(-3, 0)$
$C(3, 3) \to C'(-3, 3)$
Final Answer: $A'(0, 0), B'(-3, 0), C'(-3, 3)$
Q4. Complete the table showing rotation coordinates for point $(x, y)$:
| Rotation Angle & Direction | Coordinate Transformation Rule |
|---|---|
| $90^\circ$ Clockwise (CW) | $(x, y) \to (y, -x)$ |
| $90^\circ$ Counter-Clockwise (CCW) | $(x, y) \to (-y, x)$ |
| $180^\circ$ (Half-turn CW or CCW) | $(x, y) \to (-x, -y)$ |
| $270^\circ$ Clockwise | $(x, y) \to (-y, x)$ |
| $360^\circ$ (Full turn) | $(x, y) \to (x, y)$ |
Q5. (a) Draw $\Delta DEF$ with vertices $D(1, 1), E(3, 1), F(2, 3)$. (b) Enlarge $\Delta DEF$ by scale factor 2 with origin as centre. (c) What is the scale factor?
Step 1: Dilation rule for $k=2$: $(x, y) \to (2x, 2y)$.
$D(1, 1) \to D'(2, 2)$
$E(3, 1) \to E'(6, 2)$
$F(2, 3) \to F'(4, 6)$
Step 2: Scale factor: $k = \frac{E'D'}{ED} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$.
Final Answer: Enlarged vertices: $D'(2, 2), E'(6, 2), F'(4, 6)$ | Scale Factor $k = 2$
Q6. A rectangle with length $12\text{ cm}$ and width $8\text{ cm}$ is enlarged by a scale factor $k = 6$. Find the dimensions and area of the image.
Step 1: Calculate new dimensions:
$\text{Length}' = 12 \times 6 = 72\text{ cm}$
$\text{Width}' = 8 \times 6 = 48\text{ cm}$
Step 2: Calculate Area:
$\text{Area}' = 72 \times 48 = 3456\text{ cm}^2$ (or $\text{Area}' = \text{Original Area} \times k^2 = 96 \times 36 = 3456\text{ cm}^2$).
Final Answer: Length = $72\text{ cm}$, Width = $48\text{ cm}$, Area = $3456\text{ cm}^2$
Q7. A rectangle has length $10\text{ cm}$ and width $6\text{ cm}$. Find the dimensions of the image if it is reduced by a scale factor $k = \frac{1}{2}$.
Step 1: Compute reduced dimensions:
$\text{Length}' = 10 \times \frac{1}{2} = 5\text{ cm}$
$\text{Width}' = 6 \times \frac{1}{2} = 3\text{ cm}$
$\text{Area}' = 5 \times 3 = 15\text{ cm}^2$
Final Answer: Length = $5\text{ cm}$, Width = $3\text{ cm}$, Area = $15\text{ cm}^2$
Q8. Triangle ABC has vertices $A(1, 1), B(3, 1), C(2, 3)$. Find the vertices of image $\Delta A'B'C'$ when enlarged by scale factor $k = 3$ with centre at origin.
Rule: $(x, y) \to (3x, 3y)$:
$A(1, 1) \to A'(3, 3)$
$B(3, 1) \to B'(9, 3)$
$C(2, 3) \to C'(6, 9)$
Final Answer: $A'(3, 3), B'(9, 3), C'(6, 9)$
Review Exercise 8 • Official Board MCQs & Review Problems
Page 145Q1. Choose the correct option for each of the following (12 Board MCQs):
Rule: $(-3 + 3, 7 - 4) = (0, 3)$. Correct: (0, 3)
Rule: $(0 + h, 0 + k) = (h, k)$. Correct: (h, k)
Rule: $(x, -y) \implies (-3, -4)$. Correct: (-3, -4)
Rule: $(-0, 0) = (0, 0)$. Correct: (0, 0)
Standard rule: Correct: (x, -y)
Standard rule: Correct: (-x, y)
Rule: Correct: (-y, x)
Rule: Correct: (y, -x)
Rule: $(kx, ky) \implies (\frac{1}{2}x, \frac{1}{2}y)$. Correct: ($\frac{1}{2}x, \frac{1}{2}y$)
Side lengths remain unchanged. Correct: no change
Isometries preserve distances and area. Correct: congruent
Dilations preserve angle measures and proportional shapes. Correct: similar
Q2. Find the image of the point: (i) $(-2, 3)$ when translated 4 units right and 2 units up. (ii) $(5, -1)$ when translated 3 units left and 5 units down.
(i) $(x + 4, y + 2) \implies (-2 + 4, 3 + 2) = (2, 5)$.
(ii) $(x - 3, y - 5) \implies (5 - 3, -1 - 5) = (2, -6)$.
Final Answer: (i) $(2, 5)$ | (ii) $(2, -6)$
Q3. Find the reflection of point: (i) $(6, -4)$ across x-axis. (ii) $(-3, -8)$ across y-axis.
(i) Across x-axis $(x, -y)$: $(6, -(-4)) = (6, 4)$.
(ii) Across y-axis $(-x, y)$: $(-(-3), -8) = (3, -8)$.
Final Answer: (i) $(6, 4)$ | (ii) $(3, -8)$
Q4. Triangle PQR has vertices $P(1, 2), Q(4, 2), R(2, 5)$. Find the image coordinates after a $90^\circ$ anti-clockwise rotation about origin.
Rule $(x, y) \to (-y, x)$:
$P(1, 2) \to P'(-2, 1)$
$Q(4, 2) \to Q'(-2, 4)$
$R(2, 5) \to R'(-5, 2)$
Final Answer: $P'(-2, 1), Q'(-2, 4), R'(-5, 2)$
Q5. A line segment AB has length $8\text{ cm}$. Find the length of its image under dilation with scale factor: (i) $k = 2$ (ii) $k = \frac{1}{2}$.
(i) Enlargement $k = 2$: $\text{Length}' = 8 \times 2 = 16\text{ cm}$.
(ii) Reduction $k = \frac{1}{2}$: $\text{Length}' = 8 \times \frac{1}{2} = 4\text{ cm}$.
Final Answer: (i) $16\text{ cm}$ | (ii) $4\text{ cm}$
🎯 Unit Synthesis Summary
Geometric transformations systematically reposition or resize figures on the Cartesian plane. Translations, reflections, and rotations are rigid transformations (isometries) because they preserve side lengths, interior angles, and area (yielding congruent figures $\cong$). Dilations are non-rigid transformations that change dimensions proportionally by scale factor $k$ while preserving angles (yielding similar figures $\sim$).
More Chapter Notes for Class 8 (FBISE)
MathematicsTest Your Knowledge on Chapter 8: Class 8 Mathematics - Ch 8: Geometric Transformations Mastery Guide (FBISE)
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