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1. Translations and rigid transformations

Warm-up

Three quick questions from earlier lessons.

Discover

A triangle slides 3 units right and 2 units down without turning. What happens to its side lengths?

They stay the same
They grow by 3

Key idea

A translation moves every point by the same directed distance.

A translation by

(a, b)

sends

(x, y)

(x + a, y + b)

. Translations, reflections and rotations are rigid transformations: they preserve lengths and angle measures.

Worked example

Translate $A(2, -1)$ by 4 left and 3 up.

Step 1Start at A(2, −1).
Step 2a = −4: move 4 to the left.
Step 3b = 3: move 3 up.
Step 4The image is A′(2 − 4, −1 + 3) = (−2, 2).

a = -4

b = 3

Left is negative x.

A' = (2 - 4, -1 + 3)

Add.

A' = (-2, 2)

Compute.

Your turn, guided

Your turn: the y-coordinate of $(5, 1)$ translated 6 down.

-5

1 − 6.

Practice ★

Translate the point $A(-4, -4)$ by 2 units right and 1 units down. Tap its image $A'$ .
Translate the point $A(1, 0)$ by 4 units right and 4 units down. Tap its image $A'$ .
Translate the point $A(2, -1)$ by 1 units right and 2 units down. Tap its image $A'$ .

Watch out!

Practice ★★

A translation takes $(-3, 0)$ to $(0, 5)$ . By how many units does it move points horizontally (positive = right)?
A translation takes $(-2, 4)$ to $(-4, 4)$ . By how many units does it move points horizontally (positive = right)?

A polygon with 9 sides is translated. What changes?
1. Only its position: lengths and angles stay the same
2. Its side lengths
3. Its angles

Translate triangle ABC 2 right and 2 down. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

Translate the point $A(-4, -4)$ by 2 units right and 1 units down. Tap its image $A'$ .
Translate the point $A(1, 0)$ by 4 units right and 4 units down. Tap its image $A'$ .
Translate the point $A(2, -1)$ by 1 units right and 2 units down. Tap its image $A'$ .

Summary card

Translations

$(x, y) \to (x + a, y + b)$ ; rigid: lengths and angles kept.

Watch out: left and down are negative.

2. Reflections in the coordinate plane

Warm-up

Three quick questions from earlier lessons.

Discover

A point is 3 units above the $x$ -axis. Its mirror image across the $x$ -axis is…

3 units below the axis, straight down
3 units to the left

Key idea

The line of reflection is the perpendicular bisector of each segment joining a point to its image.

Across the

x

-axis:

(x, y) \to (x, -y)

. Across the

y

-axis:

(x, y) \to (-x, y)

. Across

y = x

(x, y) \to (y, x)

Worked example

Reflect $B(-2, 5)$ in three lines.

Step 1Triangle ABC above the x-axis.
Step 2From each vertex, go straight across the axis, the same distance on the other side.
Step 3Join the images: (x, y) → (x, −y).

x

-axis:

(-2, -5)

Negate y.

y

-axis:

(2, 5)

Negate x.

y = x

(5, -2)

Swap.

Your turn, guided

Your turn: the x-coordinate of $(4, -3)$ reflected across the $y$ -axis.

-4

Negate x.

Practice ★

Reflect $A(3, 2)$ across the $x$ -axis. Tap its image.
Reflect $A(5, 6)$ across the $x$ -axis. Tap its image.
Reflect $A(-4, 1)$ across the $x$ -axis. Tap its image.

Watch out!

Practice ★★

Reflect $A(5, -1)$ across the line $y = x$ . Tap its image.
Reflect $A(-5, -3)$ across the line $y = x$ . Tap its image.

A point is 5 units from a line of reflection. How far is it from its image?

Reflect triangle ABC across the $y$ -axis. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

Reflect $A(3, 2)$ across the $x$ -axis. Tap its image.
Reflect $A(5, 6)$ across the $x$ -axis. Tap its image.
Reflect $A(-4, 1)$ across the $x$ -axis. Tap its image.

Summary card

Reflections

$x$ -axis: $(x, -y)$ ; $y$ -axis: $(-x, y)$ ; $y = x$ : $(y, x)$ .

Watch out: across the $x$ -axis, it is $y$ that changes sign.

3. Rotations about the origin

Warm-up

Three quick questions from earlier lessons.

Discover

Rotate the point $(1, 0)$ by $90^\circ$ counterclockwise about the origin. Where does it go?

$(0, 1)$
$(0, -1)$

Move P and change the angle. Compare OP and OQ.

They are always equal
OQ grows with the angle

Key idea

Rotations about the origin by quarter turns.

About the origin:

90^\circ

counterclockwise:

(x, y) \to (-y, x)

;

180^\circ

(x, y) \to (-x, -y)

;

90^\circ

clockwise (

270^\circ

counterclockwise):

(x, y) \to (y, -x)

Worked example

Rotate $C(3, 2)$ .

Step 1Triangle ABC.
Step 2Turn it about O: every point keeps its distance to O.
Step 3A quarter turn (90° counterclockwise): (x, y) → (−y, x).

90^\circ

ccw:

(-2, 3)

(−y, x).

180^\circ

(-3, -2)

(−x, −y).

90^\circ

cw:

(2, -3)

(y, −x).

Your turn, guided

Your turn: the y-coordinate of $(5, 1)$ after $90^\circ$ counterclockwise.

5

(−y, x) = (−1, 5).

Practice ★

Rotate $A(-2, -1)$ by $180^\circ$ about the origin. Tap its image.
Rotate $A(3, -2)$ by $180^\circ$ about the origin. Tap its image.

Rotate $A(1, 3)$ by $90^\circ$ counterclockwise about the origin. Tap its image.
Rotate $A(1, 2)$ by $90^\circ$ counterclockwise about the origin. Tap its image.

Watch out!

Practice ★★

A figure is rotated 7 times by $90^\circ$ in the same direction. What single rotation, between $0^\circ$ and $360^\circ$ , gives the same result? (degrees)

Boss problem. $A = (4, 6)$ is reflected across the $x$ -axis, then rotated $90^\circ$ counterclockwise about the origin, then translated 5 units right.

Rotate triangle ABC by 90° counterclockwise about the origin. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

Rotate $A(1, 3)$ by $90^\circ$ counterclockwise about the origin. Tap its image.
Rotate $A(1, 2)$ by $90^\circ$ counterclockwise about the origin. Tap its image.
Rotate $A(2, 6)$ by $90^\circ$ counterclockwise about the origin. Tap its image.

Summary card

Rotations about the origin

ccw 90°: $(-y, x)$ ; 180°: $(-x, -y)$ ; cw 90°: $(y, -x)$ .

Watch out: check the quadrant the image lands in.

4. Congruence and corresponding parts

Warm-up

Three quick questions from earlier lessons.

Discover

$\triangle ABC \cong \triangle DEF$ . Which angle of $\triangle DEF$ matches $\angle B$ ?

$\angle E$
$\angle F$

Key idea

Congruent figures match part for part.

Two figures are congruent if a sequence of rigid transformations maps one onto the other. Then corresponding sides and angles are equal. In

\triangle ABC \cong \triangle DEF

the order of letters gives the correspondence:

A \leftrightarrow D

B \leftrightarrow E

C \leftrightarrow F

Worked example

$\triangle PQR \cong \triangle STU$ , $PQ = 7$ , $\angle R = 40^\circ$ .

ST = 7

P↔S, Q↔T.

\angle U = 40^\circ

R↔U.

Your turn, guided

Your turn: $\triangle ABC \cong \triangle XYZ$ and $BC = 12$ . Find $YZ$ .

YZ = 12

B↔Y, C↔Z.

Practice ★

$\triangle ABC \cong \triangle DEF$ with $AB = 3$ , $BC = 11$ and $CA = 18$ . Find $DE$ .
$\triangle ABC \cong \triangle DEF$ with $AB = 9$ , $BC = 15$ and $CA = 16$ . Find $DE$ .

$\triangle KLM \cong \triangle STU$ . Which segment corresponds to $LM$ (length 5)?
1. $TU$
2. $ST$
3. $SU$

Watch out!

Practice ★★

$\triangle PQR \cong \triangle XYZ$ , with $\angle Q = 43^\circ$ and $\angle R = 40^\circ$ . Find $\angle X$ , in degrees.
$\triangle PQR \cong \triangle XYZ$ , with $\angle Q = 37^\circ$ and $\angle R = 47^\circ$ . Find $\angle X$ , in degrees.

Exit ticket

Three questions to check you have got it.

$\triangle ABC \cong \triangle DEF$ with $AB = 3$ , $BC = 11$ and $CA = 18$ . Find $DE$ .
$\triangle ABC \cong \triangle DEF$ with $AB = 9$ , $BC = 15$ and $CA = 16$ . Find $DE$ .
$\triangle ABC \cong \triangle DEF$ with $AB = 4$ , $BC = 11$ and $CA = 17$ . Find $DE$ .

Summary card

Corresponding parts

The order of letters in a congruence statement gives the matching.

Watch out: match by position, not by appearance.

5. Angle relationships: vertical angles, parallel lines, triangle sum

Warm-up

Three quick questions from earlier lessons.

Discover

Two lines cross, making four angles. One is $70^\circ$ . What is the angle opposite it?

$70^\circ$
$110^\circ$

Drag the corners of the triangle. Watch the three angles.

They always add up to 180°
It depends on the shape of the triangle

Key idea

Angle facts, proved with rigid transformations.

Vertical angles are equal. Angles on a straight line add to

180^\circ

. With a transversal across parallel lines: corresponding and alternate interior angles are equal; same-side interior angles add to

180^\circ

. The angles of a triangle add to

180^\circ

Worked example

A triangle has angles $x$ , $2x$ and $60^\circ$ .

x + 2x + 60 = 180

Angle sum.

3x = 120

x = 40

Solve.

40^\circ

80^\circ

60^\circ

The angles.

Your turn, guided

Your turn: a straight angle is split into $115^\circ$ and $y$ .

y = 65^\circ

180 − 115.

Practice ★

Two angles of a triangle measure $32^\circ$ and $20^\circ$ . Find the third angle, in degrees.
Two angles of a triangle measure $50^\circ$ and $35^\circ$ . Find the third angle, in degrees.

Two lines cross. One angle measures $30^\circ$ . Find the angle vertically opposite it, in degrees.

Watch out!

Practice ★★

A transversal crosses two parallel lines and makes a $33^\circ$ angle with the first. Find the co-interior (same-side interior) angle it makes with the second line, in degrees.
A transversal crosses two parallel lines and makes a $80^\circ$ angle with the first. Find the co-interior (same-side interior) angle it makes with the second line, in degrees.

Exit ticket

Three questions to check you have got it.

Two angles of a triangle measure $32^\circ$ and $20^\circ$ . Find the third angle, in degrees.
Two angles of a triangle measure $50^\circ$ and $35^\circ$ . Find the third angle, in degrees.
Two angles of a triangle measure $41^\circ$ and $80^\circ$ . Find the third angle, in degrees.

Summary card

Angle relationships

Vertical: equal. Straight line: 180°. Parallel lines: corresponding/alternate equal, same-side interior 180°. Triangle: 180°.

Watch out: equal or supplementary? Picture the rotation.

Answers: Rigid transformations and congruence

1. Translations and rigid transformations

$(-2, -5)$
$(5, -4)$
$(3, -3)$

$3$
$-2$

Only its position: lengths and angles stay the same

$(-3, -2),\ (0, -2),\ (-2, 0)$

$(-2, -5)$
$(5, -4)$
$(3, -3)$

2. Reflections in the coordinate plane

$(3, -2)$
$(5, -6)$
$(-4, -1)$

$(-1, 5)$
$(-3, -5)$

$10$

$(-4, 3),\ (-7, 3),\ (-5, 5)$

$(3, -2)$
$(5, -6)$
$(-4, -1)$

3. Rotations about the origin

$(2, 1)$
$(-3, 2)$

$(-3, 1)$
$(-2, 1)$

$270$

$11$

$(-3, 3),\ (-3, 6),\ (-5, 4)$

$(-3, 1)$
$(-2, 1)$
$(-6, 2)$

4. Congruence and corresponding parts

$TU$

$97$
$96$

5. Angle relationships: vertical angles, parallel lines, triangle sum

$128$
$95$

$30$

$147$
$100$

$128$
$95$
$59$