Congruent shapes and transformations

Two shapes are congruent when one can be laid on the other so that it covers it exactly.

• Congruent shapes have the same sides and the same angles: only their place is different.
• To check, trace one shape on tracing paper, and try to lay the paper on the other shape.
• Two stamps from the same sheet, or two tiles on the same kitchen floor, are congruent shapes.

The triangles $\displaystyle ABC$ and $\displaystyle DEF$ are congruent: laying $\displaystyle ABC$ on $\displaystyle DEF$, the vertex $\displaystyle A$ falls on $\displaystyle D$, $\displaystyle B$ on $\displaystyle E$, and $\displaystyle C$ on $\displaystyle F$. These are corresponding vertices.

• Corresponding sides are equal: $\displaystyle {AB}={DE}$, $\displaystyle {BC}={EF}$, $\displaystyle {CA}={FD}$.
• Corresponding angles are equal: $\displaystyle {\angle A}={\angle D}$, $\displaystyle {\angle B}={\angle E}$, $\displaystyle {\angle C}={\angle F}$.
• In the drawing, the same number of small dashes marks equal sides, and the same Greek letter equal angles.

Triangle $\displaystyle KLM$ covers triangle $\displaystyle PQR$ exactly: $\displaystyle K$ falls on $\displaystyle P$, $\displaystyle L$ on $\displaystyle Q$, and $\displaystyle M$ on $\displaystyle R$.
$\displaystyle {KL}={6}$ cm, $\displaystyle {LM}={8}$ cm, $\displaystyle {\angle M}={40^{\circ}}$.
Find:

$\displaystyle PQ$

$\displaystyle QR$

$\displaystyle \angle R$

A transformation moves a shape to a new place without changing its sides and angles. The shape in the new place is called the image of the shape, and it is congruent to it.

• The image of the point $\displaystyle A$ is written $\displaystyle A'$ (read: “A prime”).
• Three transformations: translation, reflection and rotation.
• Picture the shape on tracing paper: you slide the paper (translation), flip it over (reflection) or turn it about a pin (rotation).

A translation moves all the points of the shape the same distance in the same direction.
In the drawing: triangle $\displaystyle A'B'C'$ is the image of triangle $\displaystyle ABC$ in a translation. The segment $\displaystyle AA'$ shows the path of the vertex $\displaystyle A$.

• Every point moves 4 squares right and 2 squares up: $\displaystyle {A}={(1,1)}$ goes to $\displaystyle {A'}={(5,3)}$.
• In coordinates: add 4 to the $\displaystyle x$-coordinate and 2 to the $\displaystyle y$-coordinate. So $\displaystyle {B}={(4,1)}$ goes to $\displaystyle {B'}={(8,3)}$.
• In a translation the shape neither flips nor turns.

Triangle $\displaystyle KLM$ in the drawing is moved 3 squares right and 2 squares down.
Write the coordinates of $\displaystyle K'$, $\displaystyle L'$ and $\displaystyle M'$.

A reflection in a line flips the shape over, like a mirror or a reflection in water: every point goes to the other side of the line, at the same distance from it.
In the drawing: triangle $\displaystyle A'B'C'$ is the image of triangle $\displaystyle ABC$ in a reflection in the horizontal line at height 4.

• $\displaystyle A$ is 3 squares below the line, and $\displaystyle A'$ 3 squares above it.
• The segment $\displaystyle AA'$ is perpendicular to the line.
• $\displaystyle C$ is closer to the line: one square below it, so $\displaystyle C'$ is one square above it.
• A point on the line itself stays where it is.

Triangle $\displaystyle PQR$ is reflected in the horizontal line in the drawing, at height 3.
Copy the triangle onto the squared paper below, and draw its image.
Write the coordinates of $\displaystyle P'$, $\displaystyle Q'$ and $\displaystyle R'$.

A rotation about a point $\displaystyle O$ turns the shape, like a clock hand turning about the centre.
In the drawing: triangle $\displaystyle A'B'C'$ is the image of triangle $\displaystyle ABC$ in a rotation of $\displaystyle 90^{\circ}$ about $\displaystyle O$, anticlockwise.

• Every point stays at the same distance from $\displaystyle O$: $\displaystyle B$ is 5 squares from $\displaystyle O$, and so is $\displaystyle B'$.
• The segment $\displaystyle OB$ turned a quarter turn, $\displaystyle 90^{\circ}$, and became $\displaystyle OB'$.
• A rotation is given by its centre, its angle and its direction: clockwise or anticlockwise.
• The centre of rotation $\displaystyle O$ stays where it is.

Triangle $\displaystyle ODE$ in the drawing is rotated about $\displaystyle O$ by $\displaystyle 90^{\circ}$ anticlockwise.
Write the coordinates of $\displaystyle D'$ and $\displaystyle E'$.

In the drawing, the triangles $\displaystyle DEF$, $\displaystyle KLM$ and $\displaystyle PQR$ are images of triangle $\displaystyle ABC$: one in a translation, one in a reflection and one in a rotation.
For each of them, write which transformation makes it the image of $\displaystyle ABC$, and explain.