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Cambridge IGCSE0607

Geometrical terms

International Mathematics 0607 Chapter Notes

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Geometrical terms
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Geometrical terms notes

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In this chapter

  1. 1Points, lines and angles
  2. 2Bearings
  3. 3Similar, congruent and scale factor
  4. 4Triangles
  5. 5Special quadrilaterals
  6. 6Polygons
  7. 7Solids
  8. 8Vocabulary of a circle

1. Points, lines and angles

A point marks a position. A line is straight. A vertex is a corner, where lines or edges meet. Parallel lines never meet. Perpendicular lines meet at a right angle, which is 90°.

An acute angle is less than 90°. An obtuse angle is between 90° and 180°. A reflex angle is between 180° and 360°.

Extended only: a plane is a flat surface. The perpendicular bisector of a line cuts it in half at a right angle.

Examiner insight

“The main problem was that a significant number of candidates were unable to find the obtuse angle. Many gave a reflex angle.” “90°was not acceptable as this is not the mathematical name for the angle.”

Worked example 13 marks

Do not use a calculator.

Two angles a and b lie on a straight line. Angle a = 140^(°). Work out angle b, then write down the mathematical name of each angle.

  1. 1

    angles on a straight line add to 180^(°)

  2. 2

    b = 180 - 140 = 40^(°) [1]

  3. 3

    a is between 90^(°) and 180^(°): obtuse [1]

  4. 4

    b is less than 90^(°): acute [1]

2. Bearings

A bearing is an angle measured clockwise from north. Give it with three figures: east is 090° and a small angle such as 5° is 005°.

Stand at the point you start from. Draw a north line there. Turn clockwise to face the other point.

Examiner insight

“A few candidates gave the correct bearing by standing at Q, looking north and turning clockwise to face R, remembering to give the answer as a three-figure bearing.” “Many candidates gave 90 instead of a three-figure bearing.”

Worked example 12 marks

Do not use a calculator.

A ship sails from P to Q. The bearing of Q from P is 070°. Work out the bearing of P from Q.

  1. 1

    draw a north line at Q: it is parallel to the north line at P

  2. 2

    stand at Q, face north and turn clockwise to face P: 180^(°) + 70^(°) [1]

  3. 3

    bearing of P from Q is 250° [1]

3. Similar, congruent and scale factor

Congruent shapes are exactly the same shape and size. Similar shapes are the same shape but can be different sizes.

Matching angles in similar shapes are equal. Matching lengths are all multiplied by the same scale factor.

You are not expected to show that two shapes are congruent.

scale factor = (new length)/(original length)

Examiner insight

“The most common wrong answer was A and F which are similar polygons not congruent ones as they are different sizes. This was one of the more challenging questions because candidates mixed up the meaning of similar with congruent.”

Worked example 12 marks

Do not use a calculator.

Rectangle R is 4 cm by 6 cm. Rectangle S is similar to R. The shorter side of S is 10 cm. Work out the longer side of S.

  1. 1

    scale factor = 10 ÷ 4 = 2.5 [1]

  2. 2

    longer side = 6 × 2.5 = 15 cm [1]

4. Triangles

An equilateral triangle has three equal sides and three equal angles. An isosceles triangle has two equal sides and two equal angles. A scalene triangle has no equal sides. A right-angled triangle has one angle of 90°.

The angles of a triangle add up to 180°.

a + b + c = 180^(°)

Examiner insight

“The triangle caused more problem with candidates not noticing that the diagram indicated that all sides were equal so equilateral triangle was what was needed for the mark.” “Candidates should be reminded that ‘Not to scale’ on a diagram means that answers need to be calculated.”

Worked example 12 marks

Do not use a calculator.

Triangle T has sides of 5 cm, 5 cm and 8 cm. Triangle U has sides of 4 cm, 5 cm and 6 cm. Write down the mathematical name of each triangle.

  1. 1

    T has two equal sides: isosceles [1]

  2. 2

    U has no equal sides: scalene [1]

5. Special quadrilaterals

A rhombus has four equal sides. Its angles are not all equal. A kite has two pairs of equal adjacent sides. A trapezium has one pair of parallel sides. A parallelogram has two pairs of parallel sides.

A square and a rectangle have four right angles. A square also has four equal sides.

The angles of a quadrilateral add up to 360°.

a + b + c + d = 360^(°)

Examiner insight

“The rhombus was often named as a parallelogram. The kite was named as a diamond which is not a mathematical name.” “The majority of candidates knew that the shape was a trapezium. Others just repeated the word quadrilateral.”

Worked example 11 mark

All sides of a parallelogram are the same length. The angles of the parallelogram are not all equal. Write down the mathematical name of this parallelogram.

  1. 1

    Rhombus [1]

6. Polygons

A polygon is a closed shape with straight sides. In a regular polygon all sides and all angles are equal. Otherwise it is irregular.

A pentagon has 5 sides, a hexagon 6, an octagon 8 and a decagon 10.

At each vertex an interior angle and an exterior angle add up to 180°. The exterior angles of any polygon add up to 360°.

interior angles = (n - 2) × 180^(°)

exterior angle of a regular polygon = (360^(°))/n

Examiner insight

“There appears to be a lack of understanding of the difference between interior and exterior angles of a polygon.” “It is easier to work with the exterior angle of a polygon, 180 – 168 and then divide 360 by 12.”

Worked example 13 marks

Do not use a calculator.

A regular polygon has an exterior angle of 36°. Work out the number of sides, write down the name of the polygon and work out one interior angle.

  1. 1

    number of sides = 360 ÷ 36 = 10 [1]

  2. 2

    a polygon with 10 sides is a decagon [1]

  3. 3

    interior angle = 180 - 36 = 144^(°) [1]

7. Solids

A face is one flat surface of a solid. An edge is where two faces meet. A vertex is where edges meet.

A prism has the same cross-section all along its length. A pyramid has a polygon for its base and triangles that meet at a point. A cylinder, a cone and a sphere have a curved surface.

Extended only: a tetrahedron is a pyramid with four triangular faces. A hemisphere is half a sphere. A frustum is a cone or pyramid with the top cut off parallel to the base.

Examiner insight

“Common incorrect answers to the number of faces of the prism were 7, 5 and 6. It seems identifying the number of hidden faces caused a problem for some.”

Worked example 13 marks

Do not use a calculator.

A square-based pyramid has a square base and four triangular faces. Write down its number of faces, edges and vertices.

  1. 1

    faces: 1 square + 4 triangles = 5 [1]

  2. 2

    edges: 4 round the base + 4 sloping = 8 [1]

  3. 3

    vertices: 4 round the base + 1 at the top = 5 [1]

8. Vocabulary of a circle

The distance round a circle is its circumference. A radius (plural radii) joins the centre to the circumference. A diameter passes through the centre, so it is twice the radius. A semicircle is half a circle.

A chord is a straight line joining two points on the circumference. An arc is part of the circumference. A sector is the region between two radii and an arc. A segment is the region between a chord and an arc. A tangent touches the circle at one point and is at right angles to the radius there.

Extended only: a chord cuts the circumference into a shorter minor arc and a longer major arc.

Examiner insight

“The line AB caused some confusion with candidates not realising that this means the straight line so answers such as arc and circumference were incorrect. Some gave segment as their answer which is the area defined by the line AB and the arc of the circle.” “Candidates were expected to realise that Angle BAO is 90° as the tangent AB is at right angles to the radius, AO.”

Worked example 13 marks

A and B are points on a circle with centre O. Angle AOB = 70^(°). Name the straight line AB, then work out angle OAB.

  1. 1

    AB is a chord [1]

  2. 2

    OA and OB are radii, so triangle OAB is isosceles [1]

  3. 3

    angle OAB = (180 - 70) ÷ 2 = 55^(°) [1]

Recap

  • Acute is under 90°, obtuse is between 90° and 180°, reflex is over 180°.
  • A bearing is measured clockwise from north and has three figures.
  • Congruent shapes are identical. Similar shapes differ in size by a scale factor.
  • Equilateral, isosceles, scalene, right-angled: name a triangle by its equal sides or its right angle.
  • Rhombus: four equal sides. Kite: two pairs of equal adjacent sides. Trapezium: one pair of parallel sides.
  • Exterior angles of a polygon add up to 360°. Interior plus exterior is 180°.
  • A chord joins two points on a circle. A sector has two radii. A segment has a chord.
  • Extended only: perpendicular bisector, plane, tetrahedron, hemisphere, frustum, major and minor arc.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write down the mathematical name for an angle of 140^(°).1 mark
  2. A quadrilateral has exactly one pair of parallel sides. Write down its mathematical name.1 mark
  3. Do not use a calculator. Two angles of a triangle are 38^(°) and 52^(°). Work out the third angle. Is the triangle right-angled? Is it isosceles or scalene?3 marks
  4. Do not use a calculator. A right-angled triangle has sides 3 cm, 4 cm and 5 cm. A similar triangle has a longest side of 20 cm. Work out its other two sides.3 marks
  5. Do not use a calculator. The interior angles of a polygon add up to 1080^(°). Work out the number of sides and write down the name of the polygon.3 marks
  6. Do not use a calculator. Kite ABCD has AB = AD and CB = CD. Angle ABC = 70^(°) and angle BAD = 120^(°). Work out angle BCD.2 marks
  7. A triangular prism has two triangular ends and three rectangular faces. Write down its number of faces, edges and vertices.3 marks
  8. P and Q are points on a circle with centre O. Angle POQ = 100^(°). T is a point on the tangent at P, on the same side of OP as Q. Work out angle OPQ and angle QPT. Write down the mathematical name of the region between the line PQ and the shorter arc PQ.5 marks
  9. Do not use a calculator. A drone leaves its base P and flies on a bearing of 035° to a mast Q. It then flies on a bearing of 125° to a tower R. The distances PQ and QR are equal. (a) Work out the bearing of P from Q. Use it to work out angle PQR and write down the mathematical name of this angle. (b) Write down the two mathematical names that describe triangle PQR. (c) Work out the bearing of R from P and the bearing of P from R.6 marks
  10. Do not use a calculator. A floor tile is a regular polygon. Each interior angle is 5 times each exterior angle. (a) Work out the number of sides of the tile. (b) Two of these tiles lie flat and share one edge. A third tile is a different regular polygon with sides of the same length. It fits exactly into the gap at one end of the shared edge. Work out the interior angle of the third tile and write down its mathematical name.6 marks

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