Maths · 5 Geometry

5.1 Geometrical terms

Before you start

Measuring angles in degrees, and knowing that a full turn is $360 °$ .
Angles on a straight line add up to $180 °$ .
The four compass directions north, east, south and west, and that they are a quarter turn apart.
Multiplying and dividing by decimals such as $2.5$ and $0.5$ .

Syllabus objectives covered (3)

Use and interpret the following geometrical terms: • point • vertex • line • plane • parallel • perpendicular • perpendicular bisector • bearing • right angle • acute, obtuse and reflex angles • interior and exterior angles • similar • congruent • scale factor.
Use and interpret the vocabulary of: • triangles • special quadrilaterals • polygons • solids.
Use and interpret the vocabulary of a circle.

Part 1 of 8

Points, lines, planes and vertices

A point is an exact position. It has no length, no width and no thickness. You mark it with a small cross or dot and name it with a capital letter, such as $A$ .
A line is straight, has no thickness and goes on forever in both directions. The straight piece between two points $A$ and $B$ is the line segment $A B$ . If a question asks for the length of 'the line $A B$ ', it means the segment from $A$ to $B$ , because a line that goes on forever has no length.
A plane is a flat surface that goes on forever in every direction. A sheet of paper, or one face of a cube, is a piece of a plane. A curved surface, such as the side of a tin can, is not part of a plane.
A vertex is a point where lines or edges meet: a corner. A triangle has three vertices and a cube has eight. The plural of vertex is vertices.

Part 2 of 8

Parallel, perpendicular and the perpendicular bisector

Two lines are parallel if they go in exactly the same direction. They stay the same distance apart, so they never meet, however far you extend them. On a diagram, matching arrowheads drawn on two lines mark them as parallel.
In three dimensions, never meeting is not enough. On a cuboid, take a top edge running left to right and a bottom edge running front to back. They never meet, but they are not parallel, because they do not go the same way. Parallel means same direction.
Two lines are perpendicular if they are at a right angle, $90 °$ , to each other. A small square drawn in the corner where they meet marks the right angle. The $x$ -axis and $y$ -axis of a graph are perpendicular.
To bisect something means to cut it into two equal parts. The perpendicular bisector of a line segment $A B$ is the line that passes through the midpoint of $A B$ and is perpendicular to $A B$ . The name carries both facts: it halves $A B$ , and it crosses $A B$ at $90 °$ .
Every point on the perpendicular bisector of $A B$ is the same distance from $A$ as it is from $B$ . So if $P$ is on it and $P A = 7$ cm, then $P B = 7$ cm, with no measuring needed.
graph diagram — not drawn yet
A horizontal line segment AB with its midpoint M marked and a single tick on each of AM and MB, a vertical line through M with a right-angle square at M, a point P on the vertical line above M, and dashed lines PA and PB each labelled 7 cm.
The line through M halves AB at a right angle, and P on it is 7 cm from both A and B.

Part 3 of 8

Naming angles by size, and interior and exterior angles

An angle measures an amount of turn. A full turn is $360 °$ , and half a turn, which makes a straight line, is $180 °$ . The names acute, right, obtuse and reflex depend only on the size of the angle.
Acute: more than $0 °$ and less than $90 °$ . Right angle: exactly $90 °$ . Obtuse: more than $90 °$ and less than $180 °$ . Reflex: more than $180 °$ and less than $360 °$ .
graph diagram — not drawn yet
Four angles drawn side by side, each made of two rays from a vertex with the angle marked by an arc: 40° labelled acute, 90° marked with a small square and labelled right angle, 130° labelled obtuse, and 230° with the arc going the long way round labelled reflex.
The boundaries are 90° and 180°; a reflex angle is the long way round.
The boundaries matter. $90 °$ is a right angle, so it is neither acute nor obtuse. $180 °$ is a straight angle, so it is neither obtuse nor reflex. That makes $179 °$ obtuse but $181 °$ reflex.
Where two lines meet at a point, there are two angles between them, and together they make a full turn of $360 °$ . If the small one is $130 °$ , the other, going the long way round, is $360 ° - 130 ° = 230 °$ . When a question asks for the reflex angle, it wants the large one.
An interior angle of a polygon is the angle inside the shape at a vertex, between the two sides that meet there.
For an exterior angle, extend one side straight on past the vertex. The exterior angle is the angle between that extension and the next side, outside the shape.
graph diagram — not drawn yet
An irregular convex pentagon with one side extended as a dashed line past a vertex Q; the interior angle at Q is marked 132° inside the shape and the exterior angle between the dashed extension and the next side is marked 48° outside the shape.
The interior and exterior angles at Q lie along one straight line, so they add up to 180°.
At a vertex where the interior angle is less than $180 °$ , the interior and exterior angles sit side by side along a straight line. Angles on a straight line add up to $180 °$ , so
$exterior angle = 180^{\circ} - interior angle$
Example: an interior angle of $132 °$ has an exterior angle of $180 ° - 132 ° = 48 °$ . The exterior angle is how far you turn at that corner if you walk round the edge of the shape, so a sharp corner (small interior angle) means a big turn.

Part 4 of 8

Bearings

A bearing gives a direction as an angle. It is always measured from north, it always turns clockwise, and it is always written with three figures.
'The bearing of $B$ from $A$ ' means: stand at $A$ , face north, and turn clockwise until you face $B$ . The angle you turn through is the bearing. The point named after the word 'from' is where you stand and draw the north line.
graph diagram — not drawn yet
A point A with a vertical arrow upwards labelled N, a line from A to a point B up and to the right, and a clockwise arc from the north arrow to AB labelled 040°.
The bearing of B from A starts at the north line at A and turns clockwise to B.
Three figures means you put zeros in front of a small angle. An angle of $40 °$ clockwise from north is written as the bearing $040 °$ , and $7 °$ is written $007 °$ .
The compass points have fixed bearings: north $000 °$ , east $090 °$ , south $180 °$ , west $270 °$ . Halfway between them: north-east $045 °$ , south-east $135 °$ , south-west $225 °$ , north-west $315 °$ .
Bearings run from $000 °$ to $360 °$ . Any direction on the west side of the north–south line has a bearing between $180 °$ and $360 °$ , so the clockwise turn from north to reach it is a reflex angle.
Working out the bearing of $A$ from $B$ when you are given the bearing of $B$ from $A$ belongs to subtopic 5.2. Here the skill is knowing exactly what a bearing means and writing it correctly.

Part 5 of 8

Congruent, similar and scale factor

Two shapes are congruent if they are exactly the same shape and the same size. One would fit exactly on top of the other, although you might have to turn it round or flip it over first. Matching sides are equal and matching angles are equal.
Two shapes are similar if they are the same shape but not necessarily the same size. One is an enlargement of the other, possibly turned or flipped as well. Matching angles are equal, and every length has been multiplied by the same number.
That number is the scale factor. Divide a length on the new shape by the matching length on the original shape.
$scale factor = \frac{length on the new shape}{matching length on the original shape}$
Match the sides before you divide: shortest with shortest, longest with longest. Every length is multiplied by the same number, so the order of the side lengths is the same in both shapes. Dividing a short side by a long side gives a wrong scale factor.
A scale factor greater than 1 makes the shape bigger. A scale factor between 0 and 1 makes it smaller: scale factor $0.5$ halves every length. A scale factor of exactly 1 changes nothing, so the two shapes are congruent.
Angles do not change when a shape is enlarged. A similar triangle three times as long has exactly the same three angles. Only the lengths are multiplied.
The trap: adding the same amount to every side does not give a similar shape. Add $3$ cm to each side of a $3$ cm by $5$ cm rectangle and you get $6$ cm by $8$ cm. But $6 \div 3 = 2$ and $8 \div 5 = 1.6$ , so the lengths were not multiplied by the same number, and the shapes are not similar.

Part 6 of 8

Triangles and special quadrilaterals

Triangles are named by their sides. Equilateral: all three sides equal, and all three angles are $60 °$ . Isosceles: two sides equal, and the two angles opposite those sides are equal. Scalene: no two sides equal, and no two angles equal.
A right-angled triangle has one angle of $90 °$ . That describes an angle, not the sides, so a right-angled triangle can also be isosceles (angles $90 °$ , $45 °$ , $45 °$ ) or scalene.
On diagrams, sides with the same number of small dashes (tick marks) are equal, and sides with matching arrowheads are parallel. These marks are information: read them before you name a shape.
Parallelogram: both pairs of opposite sides are parallel and equal. Opposite angles are equal. The diagonals cut each other in half, which is called bisecting each other.
Rectangle: a parallelogram with four right angles. Its diagonals are equal in length and bisect each other.
Rhombus: a parallelogram with all four sides equal. Its diagonals bisect each other at right angles.
Square: four equal sides and four right angles. A square is a special rectangle and a special rhombus, so it has every property of both. Its diagonals are equal and bisect each other at right angles.
Kite: two pairs of equal sides, where the two sides in each pair are next to each other (adjacent), not opposite. The diagonals cross at right angles. The diagonal joining the two vertices where equal sides meet cuts the other diagonal in half. The two angles where unequal sides meet are equal to each other.
Trapezium: one pair of parallel sides. The other two sides are not parallel to each other.
graph diagram — not drawn yet
Six quadrilaterals in two rows, each labelled: square, rectangle, parallelogram, rhombus, kite and trapezium, with tick marks showing equal sides, matching arrowheads showing parallel sides, small squares at right angles, and the diagonals of the rhombus and the kite drawn dashed with a right-angle mark where they cross.
The tick marks and arrowheads carry the definitions; the kite's equal sides are adjacent, the rhombus's are all four.
The kite trap: in a kite the equal sides are next to each other. If a quadrilateral has both pairs of opposite sides equal, it is a parallelogram, not a kite.

Part 7 of 8

Polygons and solids

A polygon is a flat, closed shape with straight sides. It is named by its number of sides: a pentagon has 5, a hexagon 6, an octagon 8 and a decagon 10.
A polygon is regular if all its sides are equal and all its angles are equal. Both conditions are needed. If either one fails, the polygon is irregular.
A rhombus that is not a square has four equal sides but two different sizes of angle, so it is irregular. A rectangle that is not a square has four equal angles but sides of two different lengths, so it is irregular too. The only regular quadrilateral is the square.
A solid is described by its parts. A face is a flat surface. An edge is a line where two surfaces meet. A vertex is a corner where edges meet. Surface is the general word for any outside part of a solid, flat or curved.
Cube: six square faces. Cuboid: six rectangular faces. Each has 12 edges and 8 vertices.
A prism has the same cross-section all along its length. Its two end faces are identical, parallel polygons, joined by flat side faces (rectangles, when the length runs at right angles to the ends). It is named after its end: a triangular prism has a triangle at each end.
Counting for a prism whose end has $n$ sides: $n + 2$ faces, $3 n$ edges and $2 n$ vertices. The $3 n$ edges are $n$ round each end plus $n$ along the length. A triangular prism ( $n = 3$ ) has 5 faces, 9 edges and 6 vertices.
A pyramid has a polygon base and triangular faces that meet at one vertex at the top, called the apex. A square-based pyramid has 5 faces, 8 edges and 5 vertices. A tetrahedron is a pyramid on a triangular base: 4 triangular faces, 6 edges and 4 vertices.
Some solids have curved surfaces. A cylinder has two flat circular faces joined by one curved surface. A cone has one flat circular face and a curved surface that narrows to a single vertex. A sphere has one curved surface, and every point on it is the same distance from the centre. A hemisphere is half a sphere: one flat circular face and one curved surface.
A frustum is what is left when the top of a cone or pyramid is cut off by a plane parallel to the base. A bucket with sloping sides is the shape of a frustum of a cone.
graph diagram — not drawn yet
A cone drawn with a dashed horizontal plane cutting it partway up parallel to the base; beside it the lower part alone, labelled frustum, showing its large circular base, smaller circular top and the curved surface between them.
The cut is parallel to the base, so the top of the frustum is a smaller circle.
A check for any prism, pyramid or frustum of a pyramid: faces + vertices − edges = 2. For a cube, $6 + 8 - 12 = 2$ . The syllabus does not ask you for this rule, but it catches a miscount.

Part 8 of 8

The parts of a circle

The centre is the point in the middle. Every point on the circle is the same distance from the centre, and that distance is the radius. The plural of radius is radii.
A diameter is a straight line from one side of the circle to the other, passing through the centre. It is two radii end to end, so a circle of radius $6$ cm has diameter $12$ cm.
$d = 2 r$
The circumference is the curved edge of the circle, all the way round. The same word is used for its length, which is the perimeter of the circle.
A chord is a straight line segment joining two points on the circumference. A diameter is the chord that passes through the centre, and it is the longest chord.
A tangent is a straight line that touches the circle at exactly one point. Compare it with a chord: a chord has both ends on the circle, but a tangent meets the circle once and stays outside it everywhere else.
An arc is part of the circumference. Two points on a circle split it into two arcs: the shorter is the minor arc and the longer is the major arc. A semicircle is half a circle, cut off by a diameter, so its arc is exactly half the circumference.
A sector is the region between two radii and an arc, shaped like a slice of pizza. A segment is the region between a chord and an arc, the piece cut off by one straight cut across the circle. Each comes in a minor and a major version, using the minor or major arc.
graph diagram — not drawn yet
Two circles side by side, each with centre O. Left: two radii OA and OB with the minor sector between them shaded and labelled minor sector, the minor arc AB labelled, and a tangent line touching the circle at one point labelled tangent. Right: a chord PQ not through O with the smaller region between PQ and the minor arc shaded and labelled minor segment, and a diameter drawn through O labelled diameter.
A sector has two straight edges meeting at the centre; a segment has one straight edge, a chord.
The trap is sector against segment. Look at the straight edges of the region. Two radii meeting at the centre: sector. One chord: segment.

Worked examples

Example 1. Hexagon $A B C D E F$ has interior angles $A = 150 °$ , $B = 85 °$ , $C = 160 °$ , $D = 90 °$ , $E = 200 °$ and $F = 35 °$ . (a) State the type of each angle. (b) Calculate the exterior angles at $A$ , $B$ and $D$ . (c) Explain whether the hexagon is regular.

Show the working

(a) Compare each angle with the two boundaries, $90 °$ and $180 °$ . $B = 85 °$ and $F = 35 °$ are less than $90 °$ , so they are acute.
$A = 150 °$ and $C = 160 °$ are between $90 °$ and $180 °$ , so they are obtuse. $D = 90 °$ exactly, so it is a right angle.
$E = 200 °$ is more than $180 °$ , so it is reflex. At $E$ the shape bends inwards.
(b) At $A$ , $B$ and $D$ the interior angle is less than $180 °$ , so the exterior angle is $180 °$ minus the interior angle.
$A : 180^{\circ} - 150^{\circ} = 30^{\circ}, B : 180^{\circ} - 85^{\circ} = 95^{\circ}, D : 180^{\circ} - 90^{\circ} = 90^{\circ}$
Sense check: the sharp (acute) corner at $B$ gives a big exterior angle, and the right angle at $D$ gives a right angle outside as well.
(c) A regular polygon needs all its sides equal and all its angles equal. $A = 150 °$ and $B = 85 °$ are different, so the angle condition already fails. You do not need to know the sides.

Answer: (a) $A$ obtuse, $B$ acute, $C$ obtuse, $D$ right angle, $E$ reflex, $F$ acute. (b) $30 °$ at $A$ , $95 °$ at $B$ , $90 °$ at $D$ . (c) Not regular: its interior angles are not all equal, e.g. $A = 150 °$ but $B = 85 °$ .

Example 2. A boat $B$ is on a bearing of $040 °$ from a lighthouse $L$ . A buoy $C$ is due west of $L$ . (a) Describe how to find the direction of $B$ from $L$ . (b) Write down the bearing of $C$ from $L$ . (c) Standing at $L$ , you turn clockwise from facing $B$ to facing $C$ . Calculate the angle you turn through, and state its type. (d) Find the other angle between the lines $L B$ and $L C$ .

Show the working

(a) 'From $L$ ' means you stand at $L$ . Draw a north line at $L$ and turn clockwise from it through $40 °$ . The boat is in that direction, between north and east.
(b) West is three quarter-turns clockwise from north.
$3 \times 90^{\circ} = 270^{\circ}$
(c) Both bearings are measured clockwise from the same north line at $L$ , so the clockwise turn from the direction of $B$ to the direction of $C$ is the difference between them.
$270^{\circ} - 40^{\circ} = 230^{\circ}$
$230 °$ is more than $180 °$ , so it is a reflex angle.
(d) The two angles between two lines from the same point make a full turn.
$360^{\circ} - 230^{\circ} = 130^{\circ}$
Check with a sketch: north up, $B$ a little to the right of north, $C$ straight out to the left. The smaller angle between them, measured anticlockwise from $L B$ through north to $L C$ , is $40 ° + 90 ° = 130 °$ , which agrees.

Answer: (a) Face north at $L$ and turn $40 °$ clockwise. (b) $270 °$ (c) $230 °$ , a reflex angle. (d) $130 °$ , an obtuse angle.

Example 3. Rectangle $P$ is $4$ cm by $6$ cm. Rectangle $Q$ is $10$ cm by $15$ cm. Rectangle $R$ is $6$ cm by $8$ cm. (a) Explain why $Q$ is similar to $P$ , and give the scale factor from $P$ to $Q$ . (b) Explain whether $R$ is similar to $P$ . (c) Rectangle $S$ is similar to $P$ , with scale factor $0.5$ . Find the length and width of $S$ .

Show the working

(a) Every rectangle has four right angles, so the angles already match. Whether the rectangles are similar depends only on the sides: both lengths must be multiplied by the same number.
Match the short side with the short side, and the long side with the long side, then divide new by original.
$10 \div 4 = 2.5, 15 \div 6 = 2.5$
Both multipliers are $2.5$ , so $Q$ is an enlargement of $P$ with scale factor $2.5$ .
(b) Do the same for $R$ .
$6 \div 4 = 1.5, 8 \div 6 = 1.33 \dots$
The multipliers are different, so $R$ is not similar to $P$ . Each side of $P$ has had $2$ cm added, and adding changes the shape: $R$ is closer to a square than $P$ is.
(c) Multiply each length of $P$ by the scale factor.
$4 \times 0.5 = 2, 6 \times 0.5 = 3$
A scale factor less than 1 should make the shape smaller, and it has.

Answer: (a) $10 \div 4 = 2.5$ and $15 \div 6 = 2.5$ : both sides are multiplied by the same number, so $Q$ is similar to $P$ with scale factor $2.5$ . (b) Not similar: $6 \div 4 = 1.5$ but $8 \div 6 \approx 1.33$ . (c) $2$ cm by $3$ cm.

Example 4. A round cake is a cylinder with radius $12$ cm. (a) Describe the faces and surfaces of the cake. (b) Write down the diameter of the top of the cake. (c) Two straight cuts are made from the centre of the top out to its edge, and the slice between them, which is less than half the cake, is lifted out. Name the shape of the top of the slice, and its curved edge. (d) On a second cake, one straight cut is made right across the top, not through the centre. Name the line of the cut, and the shape of the top of the smaller piece.

Show the working

(a) A cylinder has a flat circular face at each end, here the top and the bottom, joined by one curved surface around the side.
(b) A diameter is two radii end to end.
$2 \times 12 = 24$
(c) Each cut runs from the centre to the circumference, so each cut is a radius. The region between two radii and an arc is a sector. The curved edge of the slice is an arc.
The slice is less than half the cake, so it uses the shorter arc: its top is a minor sector.
(d) The cut is a straight line joining two points on the circumference, so it is a chord. The region between a chord and an arc is a segment.
The cut does not pass through the centre, so the smaller piece is the one on the far side of the chord from the centre. Its top is the minor segment.
Check by counting straight edges: the slice in (c) has two straight edges meeting at the centre, a sector. The piece in (d) has one straight edge, a segment.

Answer: (a) Two flat circular faces (top and bottom) joined by one curved surface. (b) $24$ cm. (c) The top is a minor sector, and its curved edge is an arc. (d) The cut is a chord, and the top of the smaller piece is the minor segment.

Where people go wrong

Mistake: Calling the region between a chord and an arc a sector.

Instead: Sector: two radii and an arc. Segment: one chord and an arc.

Look at the straight edges. Both straight edges of a sector run to the centre. A segment has only one straight edge, and it does not have to go near the centre.

Mistake: Giving the smaller angle when a question asks for a reflex angle, e.g. $130 °$ instead of $230 °$ .

Instead: A reflex angle is between $180 °$ and $360 °$ , so subtract the smaller angle from $360 °$ .

Two lines from a point make two angles that add up to $360 °$ . The word reflex tells you which of the two is meant.

Mistake: Saying a polygon is regular because all its sides are equal.

Instead: Regular needs equal sides and equal angles.

A rhombus has four equal sides, but a rhombus that is not a square has two different sizes of angle, so it is irregular.

Mistake: Making a similar shape by adding the same length to every side.

Instead: Multiply every length by the same scale factor.

Adding $2$ cm to each side of a $4$ cm by $6$ cm rectangle gives $6$ cm by $8$ cm, and $6 \div 4 = 1.5$ while $8 \div 6 \approx 1.33$ . The multipliers differ, so the shape has changed.

Mistake: Writing a bearing as $45 °$ , or measuring it anticlockwise, or from east.

Instead: Start at north, turn clockwise, and write three figures: $045 °$ .

A bearing is defined as a clockwise angle from north. Starting anywhere else, or turning the other way, points to a different direction.

Check yourself

State the type of each angle: $12 °$ , $90 °$ , $179 °$ , $181 °$ .
Answer
$12 °$ acute; $90 °$ right angle; $179 °$ obtuse; $181 °$ reflex, because it is more than $180 °$ .
The interior angle at one vertex of a polygon is $68 °$ . Calculate the exterior angle at that vertex.
Answer
$180 ° - 68 ° = 112 °$ .
Write down the bearing of a point that is due south-west of you.
Answer
$225 °$ . South-west is halfway between south ( $180 °$ ) and west ( $270 °$ ).
$P$ lies on the perpendicular bisector of the line segment $A B$ , and $P A = 7$ cm. Write down the length $P B$ , and give the reason.
Answer
$P B = 7$ cm. Every point on the perpendicular bisector of $A B$ is the same distance from $A$ as from $B$ .
A rectangle $2$ cm by $5$ cm is enlarged with scale factor $3$ . Write down the size of the new rectangle. Is it similar or congruent to the original?
Answer
$6$ cm by $15$ cm. It is similar (same shape, every length multiplied by $3$ ) but not congruent, because it is a different size.
A quadrilateral has all four sides equal and its diagonals cross at right angles, but none of its angles is a right angle. Write down its name.
Answer
A rhombus. A square also has four equal sides and diagonals that cross at right angles, but a square has four right angles.
Write down the number of faces, edges and vertices of a hexagonal prism.
Answer
The end has $n = 6$ sides, so $6 + 2 = 8$ faces, $3 \times 6 = 18$ edges and $2 \times 6 = 12$ vertices. Check: $8 + 12 - 18 = 2$ .
In a circle, write down the name of (a) a chord that passes through the centre, (b) the region between a chord and the minor arc.
Answer
(a) A diameter. (b) The minor segment.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Pentagon $P Q R S T$ has interior angles $P = 104 °$ , $Q = 124 °$ , $R = 75 °$ , $S = 207 °$ and $T = 30 °$ .

(a) Write down the letter of the vertex where the interior angle is reflex. 1 mark No calculator

Mark scheme

B1 S cao

(b) Write down the letters of the two vertices where the interior angle is acute. 1 mark No calculator

Mark scheme

B1 R and T, both required and no others

(c) Calculate the exterior angle at $Q$ . 1 mark No calculator

Mark scheme

B1 56[°]

(d) Explain why $P Q R S T$ is not a regular pentagon. 1 mark No calculator

Mark scheme

B1 the interior angles are not all equal, e.g. P = 104° and Q = 124° oe

Question 2. Triangle $A$ has sides $4$ cm, $6$ cm and $7$ cm. Triangle $B$ has sides $10$ cm, $15$ cm and $17.5$ cm. Triangle $C$ has sides $7$ cm, $4$ cm and $6$ cm.

(a) Write down which triangle is congruent to triangle $A$ . 1 mark No calculator

Mark scheme

B1 C

(b) Triangle $B$ is similar to triangle $A$ . Calculate the scale factor of the enlargement that maps triangle $A$ onto triangle $B$ . 1 mark No calculator

Mark scheme

B1 2.5 oe

(c) Triangle $D$ is similar to triangle $A$ . The shortest side of triangle $D$ is $6$ cm. Calculate the length of the longest side of triangle $D$ . 2 marks No calculator

Mark scheme

M1 6 ÷ 4 [= 1.5] seen and used as a multiplier, or 7 × 6 ÷ 4 oe
A1 10.5 [cm]

Question 3. A circle has centre $O$ and radius $6$ cm. $A$ and $B$ are two points on the circumference, and the straight line $A B$ does not pass through $O$ . The straight line $m$ touches the circle at $B$ and at no other point.

(a) Write down the diameter of the circle. 1 mark No calculator

Mark scheme

B1 12 [cm]

(b) Write down the mathematical name of the line segment $A B$ . 1 mark No calculator

Mark scheme

B1 chord

(c) Write down the mathematical name of the line $m$ . 1 mark No calculator

Mark scheme

B1 tangent

(d) Region $X$ is bounded by the radii $O A$ and $O B$ and the minor arc $A B$ . Region $Y$ is bounded by the line segment $A B$ and the minor arc $A B$ . Write down the mathematical name of region $X$ and of region $Y$ . 2 marks No calculator

Mark scheme

B1 X: [minor] sector
B1 Y: [minor] segment

Question 4. This question is about solids and quadrilaterals.

(a) A solid has two triangular faces that are congruent and parallel, joined by three rectangular faces. Write down the mathematical name of this solid. 1 mark No calculator

Mark scheme

B1 triangular prism

(b) Write down the number of edges and the number of vertices of this solid. 2 marks No calculator

Mark scheme

B1 9 edges
B1 6 vertices

(c) A quadrilateral has exactly one pair of parallel sides. Write down its mathematical name. 1 mark No calculator

Mark scheme

B1 trapezium

(d) Rhombus $W X Y Z$ has no right angles. Explain why $W X Y Z$ is not a regular polygon. 1 mark No calculator

Mark scheme

B1 its angles are not all equal [although its sides are all equal] oe

How this is examined

Finding a scale factor from matching sides, and using it to tell similar from congruent: A 1- or 2-mark Calculate or Write down: shapes with their side lengths given; finding missing lengths in similar shapes is taught fully in 5.3 Similarity The scale factor from one pair of matching sides, new divided by original (e.g. $8 \div 5 = 1.6$ ), is the method; if a length is asked for, that factor multiplied onto the matching side ( $9 \times 1.6 = 14.4$ cm) is the answer. Scale factor 1 means congruent. The two sides you divide must match: shortest with shortest, longest with longest.
Naming a solid from its faces and counting its edges and vertices: Write down parts worth 1 mark each: a solid described by its faces, then its number of edges or vertices The full name that fits only that solid, such as triangular prism, and each count correct, e.g. 9 edges and 6 vertices for a triangular prism. Faces + vertices − edges = 2 is a quick check before you write.
Telling sector from segment, and chord from tangent: Write down parts worth 1 mark each, describing a line or a region of a circle and asking for its mathematical name The correct word for each: chord (straight, both ends on the circle), tangent (touches at exactly one point), sector (two radii and an arc), segment (a chord and an arc). Add minor or major when the question asks which one.
Naming an angle by its size, and finding an exterior angle: Write down or Calculate parts about the angles of a given polygon, 1 mark each Each angle placed against the boundaries: acute below $90 °$ , obtuse between $90 °$ and $180 °$ , reflex above $180 °$ . The exterior angle as $180 °$ minus the interior angle, e.g. $180 ° - 132 ° = 48 °$ .
Explaining why a polygon is not regular, or naming a quadrilateral from its properties: A 1-mark Explain, or a 1-mark Write down after a list of properties For regular: name the condition that fails, with evidence, e.g. 'the angles are not all equal: $A = 150 °$ but $B = 85 °$ '. For a quadrilateral: the one name that fits every property given, e.g. exactly one pair of parallel sides is a trapezium.
Reading and writing a bearing: A 1-mark Write down: the bearing of a point given by a compass direction, or a direction written as a three-figure bearing Three figures, measured clockwise from north at the point named after 'from': due west is $270 °$ , south-west is $225 °$ , and $40 °$ is written $040 °$ .

Words to know

Plane: A flat surface that goes on forever in every direction.
Vertex: A point where lines or edges meet; a corner. Plural vertices.
Parallel: Going in exactly the same direction, so always the same distance apart.
Perpendicular: At right angles ( $90 °$ ) to each other.
Perpendicular bisector: The line through the midpoint of a line segment, at right angles to it. Every point on it is the same distance from both ends of the segment.
Bearing: A direction given as an angle measured clockwise from north, written with three figures, e.g. $045 °$ .
Reflex angle: An angle of more than $180 °$ and less than $360 °$ .
Exterior angle: The angle between one side of a polygon extended past a vertex and the next side. Where the interior angle is less than $180 °$ , interior + exterior = $180 °$ .
Congruent: Exactly the same shape and size.
Similar: The same shape: one is an enlargement of the other, so matching angles are equal and all lengths are multiplied by the same scale factor.
Scale factor: The number every length is multiplied by in an enlargement: new length ÷ matching original length.
Kite: A quadrilateral with two pairs of equal adjacent sides. Its diagonals cross at right angles.
Trapezium: A quadrilateral with one pair of parallel sides.
Regular polygon: A polygon with all sides equal and all angles equal.
Prism: A solid with the same cross-section all along its length; its two end faces are identical parallel polygons.
Frustum: The part of a cone or pyramid left when the top is cut off by a plane parallel to the base.
Chord: A straight line segment joining two points on the circumference of a circle.
Tangent: A straight line that touches a circle at exactly one point.
Sector: The region of a circle between two radii and an arc.
Segment: The region of a circle between a chord and an arc.

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