11. Reflection (Coordinate Geometry) Previous Year Questions

(a) Plotted A(1, 3), B(1, 2), C(3, 0). [1 Mark]
(b) Reflections in x-axis: D(1, −2), E(1, −3). [1 Mark]
(c) Reflections in origin: F(−1, −3), G(−1, −2). [1 Mark]
(d) Reflections in y-axis: H(−3, 0), I(−1, 2), J(−1, 3). [1 Mark]
(e) Joining in order gives a symmetrical 10-sided polygon: Decagon (or arrowhead-star decagon). [1 Mark]

(a) Points plotted. [1 Mark]
(b) B'(−2, 1), C'(−4, −1). [1 Mark]
(c) & (d) Line BB' is y = 1. Reflection of A(0, 3) across y = 1 is A'(0, −1). [1 Mark]
(e) Diagonals AA'=4 and BB'=4 are equal and perpendicular bisectors of each other. Figure is a Square (or Rhombus). [2 Marks]

(a) Correctly plotting points $A(0, 4)$, $B(1, 1)$, $C(4, 0)$ on graph scale.1 Mark

(b) Reflection in x-axis: $(x, y) \to (x, -y)$
Image of $A(0, 4)$ is $\mathbf{E(0, -4)}$.
Image of $B(1, 1)$ is $\mathbf{D(1, -1)}$.1 Mark

(c) Reflection through Origin: $(x, y) \to (-x, -y)$
Image of $B(1, 1)$ through origin is $\mathbf{F(-1, -1)}$.1 Mark

(d) Reflection in y-axis: $(x, y) \to (-x, y)$
Image of $B(1, 1)$ is $\mathbf{H(-1, 1)}$.
Image of $C(4, 0)$ is $\mathbf{G(-4, 0)}$.1 Mark

(e) Geometrical Name Joining $A \to B \to C \to D \to E \to F \to G \to H \to A$ forms an 8-sided closed figure with symmetrical diagonal axes.
Geometrical name: Octagon (or 8-pointed / 8-sided star polygon / Star Octagon).1 Mark

(i) Points $A(-4, 0), B(-3, 2), C(0, 4), D(4, 1), E(7, 3)$ plotted accurately on graph sheet. [1 Mark]
(ii) Coordinates of reflected points across the x-axis ($M_x(x, y) = (x, -y)$):
$B'(-3, -2), \quad C'(0, -4), \quad D'(4, -1), \quad E'(7, -3)$. [1 Mark]
(iii) Join in order: $A \to B \to C \to D \to E \to E' \to D' \to C' \to B' \to A$. [1 Mark]
(iv) The figure has 9 sides and 9 vertices.
Name of closed figure: Nonagon (or Arrowhead Nonagon / 9-sided polygon). [1 Mark]

Step 1 (i) Plot original vertices on Cartesian grid: $A(0, 5), B(3, 0), C(1, 0), D(1, -5)$. [1 Mark]
Step 2 (ii) & (iii) Reflection in the Y-axis has coordinate mapping $(x, y) \xrightarrow{M_y} (-x, y)$:
$B(3, 0) \xrightarrow{M_y} \mathbf{B'(-3, 0)}$
$C(1, 0) \xrightarrow{M_y} \mathbf{C'(-1, 0)}$
$D(1, -5) \xrightarrow{M_y} \mathbf{D'(-1, -5)}$. [2 Marks]
Step 3 (iv) Join points $A \to B \to C \to D \to D' \to C' \to B' \to A$ in order.
The resulting closed polygon has 7 sides and 7 vertices ($A, B, C, D, D', C', B'$).
Name of closed figure: Heptagon (also accepted: Arrowhead or concave heptagon). [1 Mark]

Step 1 (a) Plot points: $A(2, 3)$ and $B(4, 5)$. [1 Mark]
Step 2 (b) Reflection in the x-axis has mapping $(x, y) \xrightarrow{M_x} (x, -y)$:
$\mathbf{A'(2, -3)}$ and $\mathbf{B'(4, -5)}$. [2 Marks]
Step 3 (c) Reflection of $A'(2, -3)$ in the y-axis has mapping $(x, y) \xrightarrow{M_y} (-x, y)$:
$\mathbf{A''(-2, -3)}$. [1 Mark]
Step 4 (d) Transformation mapping $A(2, 3) \to A''(-2, -3)$:
$(x, y) \to (-x, -y)$ represents a **Reflection in the Origin $(0, 0)$** (or rotation through $180^\circ$ about origin). [1 Mark]

(i) Plotting Reflection in the y-axis maps $(x, y) \to (-x, y)$. Plot vertices $A(2,2), B(2,-2), C(0,-1), D(0,1)$ and connect to form $ABCD$. [1 Mark]
(ii) Coordinates
$$A'( -2, 2 ) \quad \text{and} \quad B'( -2, -2 )$$ [1 Mark]
(iii) Invariant Points Points that lie on the axis of reflection ($y$-axis, where $x = 0$) remain unchanged under reflection.
Hence, $C(0, -1)$ and $D(0, 1)$ are invariant points. [1 Mark]
(iv) Name of Polygon In $A'B'CD$:
$A'B'$ is vertical (from $(-2, -2)$ to $(-2, 2)$, length 4 units) and $DC$ is vertical along the y-axis (from $(0, -1)$ to $(0, 1)$, length 2 units).
Since $A'B' \parallel DC$, and non-parallel sides $A'D = B'C = \sqrt{2^2 + 1^2} = \sqrt{5}$, polygon $A'B'CD$ is an Isosceles Trapezium (or Trapezium). [1 Mark]
Answers: (ii) $A'(-2,2), B'(-2,-2)$, (iii) $C(0,-1)$ and $D(0,1)$, (iv) Isosceles Trapezium.

(i) Plotting Plot points $A(0,4), B(2,3), C(1,1), D(2,0)$ on the graph. [1.5 Marks]
(ii) Coordinates of Images Reflection across y-axis maps $(x, y) \to (-x, y)$:
$$B' = \mathbf{(-2, 3)}, \quad C' = \mathbf{(-1, 1)}, \quad D' = \mathbf{(-2, 0)}$$ [1.5 Marks]
(iii) Closed Figure & Symmetry Connecting $A - B - C - D - D' - C' - B' - A$ yields a symmetrical arrowhead figure.
The line of symmetry is the y-axis, whose equation is: $$\mathbf{x = 0}$$ [2 Marks]
Answers: (ii) $B'(-2,3), C'(-1,1), D'(-2,0)$; (iii) Equation of line of symmetry is $x = 0$.