2D shapes & transformations - Year 7 Mathematics

What you will learn

classify triangles by their sides and by their angles,
name and list properties of the main quadrilaterals,
perform translations, reflections and rotations on simple shapes,
identify lines of symmetry and orders of rotational symmetry.

1. Classifying triangles

By sides:

Triangles classified by sides: equilateral (all three sides equal), isosceles (two equal), scalene (all different).

Name	Property
Equilateral	all three sides equal
Isosceles	two sides equal
Scalene	all three sides different

By angles:

Name	Property
Acute	all three angles less than $90^\circ$
Right-angled	one angle equal to $90^\circ$
Obtuse	one angle greater than $90^\circ$

2. Classifying quadrilaterals

Main quadrilateral families: parallelogram, rectangle, rhombus, square, trapezium, and kite.

Main quadrilaterals and their properties

Parallelogram

Opposite sides parallel and equal. Opposite angles equal. Diagonals bisect each other.

Rectangle

Parallelogram with all angles $90^\circ$ . Diagonals are equal.

Rhombus

Parallelogram with all sides equal. Diagonals are perpendicular and bisect each other.

Square

A rectangle and a rhombus - all sides equal, all angles $90^\circ$ , diagonals equal and perpendicular.

Trapezium

At least one pair of parallel sides. (Not necessarily a parallelogram.)

Kite

Two pairs of adjacent sides equal. One line of symmetry (through the vertex between unequal pairs). Diagonals are perpendicular.

3. Transformations

A transformation changes a shape’s position or orientation without changing its size or shape.

The three rigid transformations

Translation

Slide a shape without rotating or flipping. Described by a vector: ” $5$ right and $3$ down” or $\binom{5}{-3}$ .

Reflection

Flip across a line of reflection (the “mirror”). Each point moves to its mirror image on the opposite side, the same distance from the line.

Rotation

Turn around a fixed point (the centre of rotation). Described by an angle (e.g. $90^\circ$ ) and a direction (clockwise / anticlockwise).

Worked example 1 Translation

The point $A = (2, 3)$ is translated $4$ units right and $2$ units down. Find its image $A'$ .

A' = (2 + 4,\; 3 - 2) = (6, 1).

Worked example 2 Reflection

The point $B = (3, 2)$ is reflected in the $x$ -axis. Find its image $B'$ .

Reflection in the $x$ -axis flips the $y$ -coordinate: $(x, y) \to (x, -y)$ .

B' = (3, -2).

4. Symmetry

A line of symmetry divides a shape into two mirror halves. A shape has rotational symmetry of order $n$ if it maps onto itself $n$ times in one full turn.

Shape	Lines of symmetry	Order of rotational symmetry
Equilateral triangle	$3$	$3$
Square	$4$	$4$
Rectangle (non-square)	$2$	$2$
Parallelogram (non-rectangle)	$0$	$2$
Regular hexagon	$6$	$6$

Practice

Fluency

Tier 1: basic skills

Name a triangle with all three sides equal.
Name a triangle with exactly two sides equal.
Name a triangle with all three sides different.
Name a triangle with one $90^\circ$ angle.
Name the quadrilateral with four equal sides and four right angles.
Name the quadrilateral with opposite sides parallel but no right angles and unequal adjacent sides.
Name the quadrilateral with two pairs of adjacent equal sides.
How many lines of symmetry does an isosceles triangle have?
How many lines of symmetry does a rectangle have?
What is the order of rotational symmetry of a parallelogram?
What is the order of rotational symmetry of a square?
A point $(3, 4)$ is translated $2$ left and $5$ up. Find its image.
Reflect $(2, 5)$ in the $y$ -axis.
Reflect $(-1, 4)$ in the $x$ -axis.
Rotate $(1, 0)$ by $90^\circ$ anticlockwise about the origin.

Reasoning

Tier 2: mixed practice

In a quadrilateral the angles are $x$ , $2x$ , $100^\circ$ and $80^\circ$ . Find $x$ .
A rhombus has one diagonal $6$ cm and the other $8$ cm. Find the length of a side. (Hint: the diagonals meet at right angles.)
An isosceles triangle has a base angle of $40^\circ$ . Find its apex angle.
A kite has two unequal pairs of adjacent sides: two sides of $5$ cm and two sides of $8$ cm. What is its perimeter?
List all lines of symmetry for a regular pentagon.
The point $(3, -2)$ is reflected in the $y$ -axis, then translated $1$ unit down. What are the final coordinates?
Describe fully the single transformation that takes the point $(2, 3)$ to $(-2, 3)$ .
A parallelogram has angles $x$ , $120^\circ$ , $x$ , $120^\circ$ . Find $x$ .

Reasoning

Tier 3: explain and spot the mistake

Ida says: “every square is a rectangle”. Is Ida correct? Explain.
Is every rectangle a square? Explain.
A student writes: “a trapezium is not a parallelogram”. Explain when this is true and when it is not.
Explain why a rotation of $180^\circ$ around the origin takes $(a, b)$ to $(-a, -b)$ .

Problem solving

Tier 4: real-world problems

A window has the shape of a rectangle with an equilateral triangle on top (a “house” shape). If the rectangle is $1.2$ m by $1.8$ m and the triangle sits on top of the $1.2$ m side, what is the total perimeter?
A company logo is a parallelogram with sides $6$ cm and $9$ cm. What is its perimeter?
A garden tile is an isosceles trapezium with parallel sides $10$ cm and $6$ cm, and the two slanting sides $5$ cm each. Find its perimeter.
A kite-shaped sticker has adjacent sides of $4$ cm, $4$ cm, $7$ cm, $7$ cm. Find its perimeter.
A point $P = (1, 1)$ is rotated $90^\circ$ clockwise about the origin, then reflected in the $x$ -axis. Find the final image.

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Tier 1: basic skills

Fluency

Equilateral
Isosceles
Scalene
Right-angled
Square
Parallelogram (non-rectangle, non-rhombus)
Kite
$1$
$2$
$2$
$4$
$(1, 9)$
$(-2, 5)$
$(-1, -4)$
$(0, 1)$

Tier 2: mixed practice

Reasoning

Mixed practice

$x = 60^\circ$ . Method: $x + 2x + 100 + 80 = 360$ ; so $3x = 180$ .
$5$ cm. Method: diagonals meet at right angles and bisect each other; half-diagonals are $3$ and $4$ ; Pythagoras $\sqrt{3^2 + 4^2}$ (or use the $3$ - $4$ - $5$ triangle).
$100^\circ$ . Method: base angles are both $40^\circ$ ; apex $= 180 - 80$ .
$26$ cm. Method: $5 + 5 + 8 + 8$ .
$5$ lines of symmetry.
$(-3, -3)$ . Method: reflect gives $(-3, -2)$ ; translate gives $(-3, -3)$ .
Reflection in the $y$ -axis.
$60^\circ$ . Method: $2x + 240 = 360$ , so $x = 60$ .

Tier 3: explain and spot the mistake

Reasoning

Explain and spot the mistake

Yes, Ida is correct. A rectangle is a quadrilateral with four right angles; a square meets this (and also has all sides equal), so every square is a rectangle. The extra property “all sides equal” just makes the square a special rectangle.
No. A rectangle needs only four right angles; a square also needs four equal sides. A $3 \times 5$ rectangle has four right angles but unequal sides, so it is a rectangle but not a square.
A trapezium has at least one pair of parallel sides; a parallelogram has two pairs. Under the broad (inclusive) definition, every parallelogram is a trapezium, so the student’s claim is false. Under the “exactly one pair” definition, a parallelogram is not a trapezium, so the student is correct. Both definitions are used in textbooks.
A $180^\circ$ rotation about the origin is a half-turn: each point moves to the point on the opposite side of the origin, the same distance away. Flipping direction from the origin negates both coordinates, so $(a, b) \to (-a, -b)$ .

Tier 4: real-world problems

Problem solving

Real-world problems

$6$ m total. Method: rectangle perimeter $= 2(1.2 + 1.8) = 6$ m; minus $1.2$ m (the top edge is shared) $= 4.8$ ; plus the two $1.2$ m slanted sides of the equilateral triangle; total $= 4.8 + 2.4 = 7.2$ m. Correct answer: $7.2$ m.
$30$ cm. Method: $2(6 + 9)$ .
$26$ cm. Method: $10 + 6 + 5 + 5$ .
$22$ cm. Method: $4 + 4 + 7 + 7$ .
$(1, -1)$ . Method: rotate $(1, 1)$ by $-90^\circ$ gives $(1, -1)$ ; reflect in $x$ -axis gives $(1, 1)$ . Wait - rotating $(1,1)$ by $-90^\circ$ (clockwise) gives $(1, -1)$ . Then reflecting in the $x$ -axis flips the $y$ -coordinate, giving $(1, 1)$ . Final image: $(1, 1)$ .

Prefer paper? Print the answer key as a separate booklet: open print view ->