The Lumenary · Grades 6–8

Rigid motions, congruence and similarity

Name: ______________________

Date: ______________________

You will learn to

Describe translations, rotations by multiples of 90° and reflections in the axes using coordinates
Prove that these rigid motions keep distances, and that composites and inverses of rigid motions are rigid motions
Decide whether two figures are congruent or similar by finding a sequence of transformations
Compose transformations and identify the result; for example, two reflections in intersecting lines give a rotation
Explain how a dilation with scale factor $k$ changes lengths, angles and areas

1.
Rotate the point $(2, 5)$ by $90^\circ$ anticlockwise about the origin. Where does it land?
- A. $(-5, 2)$
- B. $(5, -2)$
- C. $(5, 2)$
- D. $(-2, -5)$
2.
Translate the point $(1, -1)$ by $(-3, 4)$ . Where does it land?
- A. $(-2, 3)$
- B. $(4, -5)$
- C. $(-3, -4)$
- D. $(3, -2)$
3.
A shape is reflected in the $x$ -axis and then in the $y$ -axis. Which single move has the same effect?
- A.Rotation by $180^\circ$ about the origin
- B.Rotation by $90^\circ$ about the origin
- C.Reflection in the line $y = x$
- D.Nothing — the shape returns to where it started
4.
A triangle has area $5\ \mathrm{cm}^2$ . It is dilated with scale factor $3$ . What is the area of the image, in $\mathrm{cm}^2$ ?
5.
Triangle $P$ has corners $(0, 0)$ , $(3, 0)$ , $(0, 4)$ . Triangle $Q$ has corners $(0, 0)$ , $(6, 0)$ , $(0, 8)$ . Which is true?
- A.They are similar but not congruent.
- B.They are congruent.
- C.They are neither similar nor congruent.
- D.They are congruent and similar.
6.
Which of these does every rigid motion keep the same? (Choose all that apply.)
- A.The lengths of segments
- B.The sizes of angles
- C.Areas
- D.Orientation (clockwise stays clockwise)
- E.The position of every point
7.
Where does the point $(1, 1)$ go if you first translate by $(2, 0)$ and then rotate by $180^\circ$ about the origin?
- A. $(-3, -1)$
- B. $(1, -1)$
- C. $(3, 1)$
- D. $(-1, -1)$
8.
Two mirror lines cross at a point $O$ at an angle of $30^\circ$ . A shape is reflected in the first line and then in the second. The result is a rotation about $O$ . By how many degrees?
9.
In the Lean file, IsIsometry f means that f keeps all distances, and g ∘ f means “first f, then g”. Which statement says that the composite of two rigid motions is a rigid motion?
- A.IsIsometry f → IsIsometry g → IsIsometry (g ∘ f)
- B.IsIsometry (g ∘ f) → IsIsometry f
- C.IsIsometry f → IsIsometry g → g ∘ f = f ∘ g
- D.IsIsometry f ∧ IsIsometry g
10.
Points are pairs (x, y). Complete then(f, g) so that it returns a new function that applies f first and then g. Use it to check that reflecting in the $x$ -axis and then in the line $y = x$ is the rotation by $90^\circ$ , and that the other order is not.
Computational — Practice

Answer key — Rigid motions, congruence and similarity

1.
A quarter turn anticlockwise sends $(x, y)$ to $(-y, x)$ , so $(2, 5) \mapsto (-5, 2)$ . Check with a sketch: $(2, 5)$ is in quadrant I, and a quarter turn anticlockwise moves it into quadrant II.
2.
$(x, y) \mapsto (x + a, y + b)$ with $(a, b) = (-3, 4)$ gives $(1 - 3, -1 + 4) = (-2, 3)$ .
3.
$(x, y) \mapsto (x, -y) \mapsto (-x, -y)$ , which is the rotation by $180^\circ$ about the origin. This agrees with Theorem 4: the axes cross at $90^\circ$ , giving a rotation by $2 \cdot 90^\circ = 180^\circ$ .
4.
Both the base and the height are multiplied by $3$ , so the area $\tfrac12 \cdot \text{base} \cdot \text{height}$ is multiplied by $3^2 = 9$ : $5 \cdot 9 = 45\ \mathrm{cm}^2$ .
5.
$Q$ 's legs are $6 = 2 \cdot 3$ and $8 = 2 \cdot 4$ , and $Q$ is exactly the image of $P$ under the dilation $(x, y) \mapsto (2x, 2y)$ . So they are similar with scale factor $2$ . They are not congruent: a rigid motion keeps lengths, and $P$ 's longest side is $5$ while $Q$ 's is $10$ .
6.
Rigid motions keep lengths (by definition), angles (Theorem 3) and therefore areas. They do not in general keep orientation — reflections reverse it — and of course they move points.
7.
Translate: $(1, 1) \mapsto (3, 1)$ . Rotate by $180^\circ$ : $(3, 1) \mapsto (-3, -1)$ . In the other order the point ends at $(1, -1)$ , so the two composites are different transformations (Theorem 5).
8.
By Theorem 4 the composite is the rotation about $O$ by $2\theta = 2 \cdot 30^\circ = 60^\circ$ (anticlockwise if $\ell_2$ is $30^\circ$ anticlockwise from $\ell_1$ ; in the other order it turns $60^\circ$ the other way).
9.
The statement is IsIsometry f → IsIsometry g → IsIsometry (g ∘ f): if f and g keep distances, so does doing f then g. Option (b) is a false converse, (c) is the false claim that order never matters, and (d) is not even about composites.
10.
```
def rot90(p):
    return (-p[1], p[0])

def reflect_x(p):
    return (p[0], -p[1])

def reflect_diag(p):
    return (p[1], p[0])

def then(f, g):
    return lambda p: g(f(p))
```
In g(f(p)) the function f acts first, which is why mathematicians write the composite as $g \circ f$ (“ $g$ after $f$ ”). With the order fixed, reflecting in the $x$ -axis then in $y = x$ sends every grid point to its quarter-turn image; the reverse order sends $(1, 0)$ to $(0, -1)$ , a quarter turn clockwise.