Isometries

Unit 9 Isometries

Key unit competence: Describe shapes transformed under an isometry: central symmetry, reflection, translation or rotation.

Key ideas

• An isometry is a transformation that keeps distances: the object and its image are congruent (same shape, same size, same area).
• Central symmetry with centre $O$: $O$ is the midpoint of $[A{A}^{\prime }]$. With centre the origin: $(x,y)\mapsto (-x,-y)$; with centre $C(a,b)$: $(x,y)\mapsto (2a-x,2b-y)$.
• Reflection in a line (mirror): the mirror is the perpendicular bisector of $[A{A}^{\prime }]$. In the x-axis $(x,y)\mapsto (x,-y)$; in the y-axis $(x,y)\mapsto (-x,y)$; in $y=x$: $(x,y)\mapsto (y,x)$; in $y=-x$: $(x,y)\mapsto (-y,-x)$. Reflection reverses orientation.
• Rotation about a centre through an angle (anticlockwise is positive). About the origin: $90^{\circ }$: $(x,y)\mapsto (-y,x)$; $180^{\circ }$: $(-x,-y)$; $-90^{\circ }$ (clockwise): $(y,-x)$.
• Translation by vector $\left(\begin{array}{c}a\\ b\end{array}\right)$: $(x,y)\mapsto (x+a,y+b)$. Every point moves the same distance in the same direction.
• Common properties: lengths, angles, areas, parallelism are kept. Invariant points: the centre (central symmetry, rotation), the points of the mirror (reflection); a non-zero translation has none.

Formulas to know

Common misconceptions

• "Rotating $(x,y)$ by $90^{\circ }$ gives $(y,x)$." No: that is the reflection in $y=x$. A $90^{\circ }$ anticlockwise rotation gives $(-y,x)$.
• "A central symmetry is a reflection in a point that changes size." No: it keeps all lengths; it is the same as a rotation of $180^{\circ }$.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Triangle $A(1,1)$, $B(4,1)$, $C(1,3)$. Find its image under reflection in the y-axis.

   Show answer

   $A^{\prime }(-1,1)$, $B^{\prime }(-4,1)$, $C^{\prime }(-1,3)$.

   Common mistake: Changing the sign of $y$ instead of $x$.

2. Draw any figure, rotate it $90^{\circ }$ about the origin and state what is kept and what changes.

   Show answer

   Lengths, angles, area and shape are kept; position and orientation (direction it faces) change.

   Common mistake: Saying the area doubles.

3. Find the image of the same triangle under a translation by $\left(\begin{array}{c}-2\\ 5\end{array}\right)$.

   Show answer

   $A^{\prime }(-1,6)$, $B^{\prime }(2,6)$, $C^{\prime }(-1,8)$.

   Common mistake: Subtracting the vector instead of adding.

4. Find the mirror line of the reflection mapping $A(1,4)$ to $A^{\prime }(5,4)$.

   Show answer

   The perpendicular bisector of $A{A}^{\prime }$: the vertical line $x=3$.

   Common mistake: Giving the line through $A$ and $A^{\prime }$.

5. Find the image of $Q(-3,2)$ under rotations about $O$ of $90^{\circ }$, $180^{\circ }$ and $270^{\circ }$ anticlockwise.

   Show answer

   $90^{\circ }$: $(-2,-3)$; $180^{\circ }$: $(3,-2)$; $270^{\circ }$: $(2,3)$.

   Common mistake: Mixing up the clockwise and anticlockwise rules.

6. $A(2,3)$ is mapped onto $A^{\prime }(3,2)$. Give two different isometries that do this.

   Show answer

   Reflection in $y=x$; translation by $\left(\begin{array}{c}1\\ -1\end{array}\right)$ (also a central symmetry with centre $(2.5,2.5)$).

   Common mistake: Believing only one transformation can map one point to another.

7. Find the image of $M(0,4)$ under the central symmetry with centre $C(-1,1)$.

   Show answer

   $(2(-1)-0,2(1)-4)=(-2,-2)$.

   Common mistake: Using $(-x,-y)$ as if the centre were the origin.

8. List two properties shared by all isometries and one property that distinguishes reflection.

   Show answer

   All keep lengths and angles (so images are congruent). Reflection reverses orientation.

   Common mistake: Thinking translations change angles.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find the image of the line $y=2x+1$ under reflection in the y-axis.	a. $y=-2x+1$
   2. Which points are invariant under reflection in a line?	b. $(0,3)$
   3. The image of $(2,-4)$ under the central symmetry with centre $O$ is:	c. $(-2,4)$
   4. Rotate $(3,2)$ through $90^{\circ }$ anticlockwise about $C(1,1)$.	d. The points on the mirror line

   Show answer

   1 → a ($y=-2x+1$) · 2 → d (The points on the mirror line) · 3 → c ($(-2,4)$) · 4 → b ($(0,3)$)

   • Step 1. Reflection in the y-axis replaces $x$ by $-x$.
     Step 2. $y=2(-x)+1=-2x+1$.
   • Points on the mirror are their own images.
   • $(x,y)\mapsto (-x,-y)$.
   • Step 1. Relative to $C$: $(3-1,2-1)=(2,1)$. A $90^{\circ }$ anticlockwise turn gives $(-1,2)$.
     Step 2. Add $C$ back: $(1-1,1+2)=(0,3)$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $(-2,6)$  ·  $(-3,2)$  ·  $(4,2)$  ·  $(3,-1)$
   1. The centre of the central symmetry mapping $A(1,2)$ to $A^{\prime }(5,-4)$ is .
   2. The image of $(2,3)$ under a rotation of $90^{\circ }$ anticlockwise about $O$ is: →
   3. Translate $(1,4)$ by $\left(\begin{array}{c}3\\ -2\end{array}\right)$. →
   Show answer

   1: $(3,-1)$ · 2: $(-3,2)$ · 3: $(4,2)$

   • The centre is the midpoint of $A{A}^{\prime }$.
   • $(x,y)\mapsto (-y,x)$.
   • $(1+3,4-2)$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Reflect $(2,5)$ in the x-axis, then rotate the image $90^{\circ }$ anticlockwise about $O$.	a. $(5,2)$
   2. A point is rotated $90^{\circ }$ anticlockwise about $O$ and lands at $(0,5)$. What was the point?	b. $(-2,-3)$
   3. The image of $(2,3)$ under a rotation of $180^{\circ }$ about $O$ is:	c. Reflection
   4. Which transformation reverses the orientation (clockwise becomes anticlockwise)?	d. $(5,0)$

   Show answer

   1 → a ($(5,2)$) · 2 → d ($(5,0)$) · 3 → b ($(-2,-3)$) · 4 → c (Reflection)

   • Step 1. Reflection in the x-axis: $(2,-5)$.
     Step 2. $90^{\circ }$ anticlockwise: $(x,y)\mapsto (-y,x)$, giving $(5,2)$.
   • $(5,0)\mapsto (0,5)$.
   • A half turn: $(x,y)\mapsto (-x,-y)$.
   • A mirror image is "turned over".

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $(2,-1)$  ·  $(-1,-1)$  ·  $(-5,-3)$  ·  $\left(\begin{array}{c}3\\ -2\end{array}\right)$
   1. Under the central symmetry with centre $C(2,1)$, the image of $(5,3)$ is: →
   2. Reflect $(2,7)$ in the line $y=3$. →
   3. $A(1,1)$ is mapped to $A^{\prime }(4,-1)$ by a translation. What is the vector? →
   Show answer

   1: $(-1,-1)$ · 2: $(2,-1)$ · 3: $\left(\begin{array}{c}3\\ -2\end{array}\right)$

   • $(2\cdot 2-5,2\cdot 1-3)$.
   • Step 1. The point is $7-3=4$ units above the line.
     Step 2. The image is 4 units below: $y=3-4=-1$; $x$ stays 2.
   • $A^{\prime }-A=(3,-2)$.

Take the Unit 9 quiz → · Back to contents