Higher Tier Shape and space revision

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2 Higher Tier Shape and space revision Contents : Angles and polygons Area Area and arc length of circles Area of triangle Volume and SA of solids Spotting P, A & V formulae Transformations Constructions Loci Similarity Congruence Pythagoras Theorem SOHCAHTOA 3D Pythag and Trig Trig of angles over 90 0 Sine rule Cosine rule Circle angle theorems Vectors

3 Angles and polygons There are 3 types of angles in regular polygons Angles at = 360 the centre No. of sides Exterior = 360 angles No. of sides Interior = e angles e c c c c c c Calculate the value of c, e and i in regular polygons with 8, 9, 10 and 12 sides Answers: 8 sides = 45 0, 45 0, sides = 40 0, 40 0, sides = 36 0, 36 0, sides = 30 0, 30 0, e e e To calculate the total interior angles of an irregular polygon divide it up into triangles from 1 corner. Then no. of x 180 i Total i = 5 x 180 = 900 0

4 Area m 9m 2m What would you do to get the area of each of these shapes? Do them step by step! 2. 7m 2m 10m 3. 4m 4. 8m 5. 3m 6m 6m 1.5m 6m

5 Area of triangle There is an alternative to the most common area of a triangle formula A = (b x h)/2 and it s to be used when there are 2 sides and the included angle available. First you need to know how to label a triangle. Use capitals for angles and lower case letters for the sides opposite to them. a C b Area = ½ ab sin C B c A The included angle = = cm Area = ½ ab sin C Area = 0.5 x 6.3 x 7 x sin 59 Area = 18.9 cm cm

6 Area and arc lengths of circles Circle Area = x r 2 Circumference = x D 4.8cm 54 0 Segment Area = Area of sector area of triangle Sector Area = x x r Arc length = x x D 360 Area sector = 54/360 x 3.14 x 4.8 x 4.8 = cm 2 Area triangle = 0.5 x 4.8 x 4.8 x sin 54 = cm 2 Area segment = = 1.54cm 2 Arc length = 54/360 x 3.14 x 9.6 = 4.52 cm

7 Volume and surface area of solids 1. Calculate the volume and surface area of a cylinder with a height of 5cm and a diameter at the end of 6cm 5 Volume = xr 2 x h = 3.14 x 3 x 3 x 5 = cm 3 r 2 D 6 5 Surface area = r 2 + r 2 +( D x h) = x3 2 + x3 2 +( x6 x 5) = = cm 2 r 2 The formulae for spheres, pyramids (where used) and cones are given in the exam. However, you need to learn how to calculate the volume and surface area of a cylinder

8 Volume and surface area of solids 2. Calculate the volume and surface area of a cone with a height of 7cm and a diameter at the end of 8cm 7 Volume = 1/3 ( xr 2 x h) = 1/3 (3.14 x 4 x 4 x 7) = cm 3 8 r L r 2 L Slant height (L) = ( ) = 65 = 8.06 cm Curved surface area = r L Total surface area = r L + r 2 = (3.14 x 4 x 8.06) + (3.14 x 4 x 4) = = cm 2

9 Volume and surface area of solids 3. Calculate the volume and surface area of a sphere with a diameter of 10cm. 5 Volume = 4/3 ( xr 3 ) = 4/3 (3.14 x 5 x 5 x 5) = cm 3 Curved surface area = 4 r 2 = 4 x 3.14 x 5 x 5 = 314 cm 2 Watch out for questions where the surface area or volume have been given and you are working backwards to find the radius.

10 Spotting P, A & V formulae Which of the following expressions could be for: (a) Perimeter (b) Area (c) Volume 4l 2 h 1 r 3 V P 4 r rh 3 A 4 r 2 h A V r( + 3) 1 d 2 4 r + ½r r + 4l 3lh 2 P A P V 4 rl r(r + l) P 4 r r 2 h 3 rl A V A V A

11 Transfromations 1. Reflection y Reflect the triangle using the line: y = x then the line: y = - x then the line: x = 1 x

12 Transfromations 2. Rotation y Describe the rotation of A to B and C to D When describing a rotation always state these 3 things: No. of degrees Direction Centre of rotation e.g. a rotation of 90 0 anti-clockwise using a centre of (0, 1) B A C x D

13 Transfromations 3. Translation What happens when we translate a shape? The shape remains the same size and shape and the same way up it just. slides. Horizontal translation Use a vector to describe a translation 3-4 Vertical translation Give the vector for the translation from.. 1. A to B 2. A to D 3. B to C 4. D to C A C D B

14 Transformations 4. Enlargement y Enlarge this shape by a scale factor of 2 using centre O Now enlarge the original shape by a scale factor of - 1 using centre O x O

15 Constructions Have a look at these constructions and work out what has been done Perpendicular bisector of a line 90 0 Triangle with 3 side lengths Bisector of an angle 60 0

16 Loci A locus is a drawing of all the points which satisfy a rule or a set of constraints. Loci is just the plural of locus. A goat is tethered to a peg in the ground at point A using a rope 1.5m long 1. Draw the locus to show all that grass he can eat A 1.5m A goat is tethered to a rail AB using a rope (with a loop on) 1.5m long 1.5m 2. Draw the locus to show all that grass he can eat A B 1.5m

17 Similarity Shapes are congruent if they are exactly the same shape and exactly the same size Shapes are similar if they are exactly the same shape but different sizes How can I spot similar triangles? These two triangles are similar because of the parallel lines Triangle C Triangle A Triangle B All of these internal triangles are similar to the big triangle because of the parallel lines

18 Similarity Triangle 2 These two triangles are similar.calculate length y y = = 8.5m Same multiplier x m 15.12m y 7.2m Triangle 1 x 2.1 Multiplier = = 2.1

19 Similarity in 2D & 3D Don t fall into the trap of thinking that the scale factor can be found by dividing one area by another area 156 cm 2 These two cylinders are similar. Calculate length L and Area A. Write down all these equations immediately: 6.2 x scale factor = L A x scale factor 2 = x scale factor 3 = scale factor 3 = /214 scale factor 3 = scale factor = 2.5 So 6.2 x 2.5 = L and A x = 156 L = 15.5cm A = 24.96cm 2 L A 6.2cm Volume = cm 3 Volume = 214cm 3

20 Shapes are congruent if they Congruence are exactly the same shape and exactly the same size There are 4 conditions under which 2 triangles are congruent: SSS - All 3 sides are the same in each triangle 18m 10m 13m 13m 10m 18m SAS - 2 sides and the included angle are the same in each triangle 11cm cm 9cm cm

21 ASA - 2 angles and the included side are the same in each triangle cm cm Be prepared to justify 52 0 these congruence rules RHS - The right angle, hypotenuse and another side are the same in each triangle by PROVING that they work 5m 5m 12m 12m

22 Pythagoras Theorem How to spot a Be prepared to leave your answer Pythagoras question in surd form (most likely in the non-calculator exam) Right angled F 45cm D Hyp 2 = a 2 + b 2 triangle Calculate the size No angles involved in question How to spot the Hypotenuse Longest side & opposite Calculating the Hypotenuse D 21cm DE of DE to 1 d.p.? 2 = DE 2 = Calculating a DEshorter 2 = 45 side DE = 45 F 6cm E ADE = 9 x 5 Calculate the size DE = 3 5 cm of DE in surd form 3cm B 11m?? 16m Calculate the size of AC to 1 d.p. C E Hyp 2 = a 2 + b 2 DE 2 = DE 2 = DE 2 = 2466 DE = 2466 DE = DE = 49.7cm Hyp 2 = a 2 + b = AC = AC = AC = AC = AC = AC AC = 11.6m

23 Pythagoras Questions Look out for the following Pythagoras questions in disguise: y x Find the distance between 2 co-ords Finding lengths in isosceles triangles x x O Finding lengths inside a circle 1 (angle in a semi -circle = 90 0 ) O Finding lengths inside a circle 2 (radius x 2 = isosc triangle)

24 SOHCAHTOA How to spot a Trigonometry question Right angled triangle An angle involved in question Label sides H, O, A Write SOHCAHTOA Write out correct rule Substitute values in If calculating angle use 2 nd func. key Calculating an angle Calculating a side B D 26cm O F 53cm Calculate the size of to 1 d.p. O 11m A H D? H A 73 0 Calculate the size of BC to 1 d.p. C E SOHCAHTOA Tan = O/A Tan = 26/53 Tan = = SOHCAHTOA Sin = O/H Sin 73 = 11/H H = 11/Sin 73 H = 11.5 m

25 3D Pythag and Trig Always work out a strategy first 1a Calculate the length of the longest diagonal inside a cylinder 2a Calculate the height of a square-based pyramid Find base diagonal 1 st 20cm 12cm Hyp 2 = Hyp 2 = Hyp 2 = 544 Hyp = 544 Hyp = 23.3 cm 11m 5m D/2 5m D 2 = D 2 = 50 D = = H = H H 2 = H = 10.4 m 1b 20cm Calculate the angle this diagonal makes with the vertical 12cm SOHCAHTOA Tan = 12/20 Tan = 0.6 = b 10.4m Calculate the angle between a sloping face and the base 2.5m SOHCAHTOA Tan = 10.4/2.5 Tan = 4.16 =

26 Trig of angles > 90 0 The Sine Curve We can use this graph to find all the angles (from 0 to 360) which satisfy the equation: Sin = 0.64 First angle is found on your calculator INV, Sin, 0.64 = You then use the symmetry of the graph to find any others. Sine 1 = and ? ? = =

27 Trig of angles > 90 0 The Cosine Curve We can use this graph to find all the angles (from 0 to 360) which satisfy the equation: Cos = Use your calculator for the 1 st angle INV, Cos, = You then use the symmetry of the graph to find any others. Cosine 1? = = = and ?

28 Trig of angles > 90 0 The Tangent Curve We can use this graph to find all the angles (from 0 to 360) which satisfy the equation: Tan = 4.1 Use your calculator for the 1 st angle INV, Tan, 4.1 = You then use the symmetry of the graph to find any others. Tangent ? -10? = = = and

29 Sine rule If there are two angles involved in the question it s a Sine rule question. Use this version of the rule to find angles: Sin A = Sin B = Sin C a b c Use this version of the rule to find sides: a = b = c. Sin A Sin B Sin C e.g. 1 e.g. 2 A b b C C c a 23m a 8m 7m B B Sin A = Sin B = Sin C a = b = c. a b c Sin A Sin B Sin C Sin = Sin B = Sin 62 7 b 23 Sin = Sin 62 x 7 23 Sin = = = b =?. Sin 9 Sin B Sin 52? = 8 x Sin 52 Sin 9?= 40.3m? c 9 0 A

30 Cosine rule If there is only one angle involved (and all 3 sides) it s a Cosine rule question. Use this version of the rule to find sides: a 2 = b 2 + c 2 2bc Cos A Use this version of the rule to find angles: Cos A = b 2 + c 2 a 2 2bc Always label the one angle involved - A e.g. 1 A 32cm? b 67 0 C 45cm a 2 = b 2 + c 2 2bc Cos A a 2 = x 32 x 45 x Cos 67 a 2 = a = cm c a B B 2.3m c Cos A = b 2 + c 2 a 2 2bc Cos = x 2.1 x 2.3 Cos = = A 3.4m a e.g m b C

31 How to tackle Higher Tier trigonometry questions Triangle in the question? Have you just got side lengths in the question? Yes Yes Is it right angled? No Are all 3 side lengths involved in the question? Yes No No Yes Use SOHCAHTOA Use the Pythagoras rule Hyp 2 = a 2 + b 2 Use this Sine rule if you are finding a side a = b = c Sin A Sin B Sin C Use this Sine rule if you are finding an angle Sin A = Sin B = Sin C a b c Use this Cosine rule if you are finding a side a 2 = b 2 + c 2 2bcCosA Label a as the side to be calculated Use this Cosine rule if you are finding an angle CosA = b 2 + c 2 a 2 2bc Label A as the angle to be calculated

32 Extra tips for trig questions Redraw triangles if they are cluttered with information or they are in a 3D diagram Right angled triangles can be easily found in squares, rectangles and isosceles triangles Remember to use the Shift Button when calculating an angle The ambiguous case only occurs for sine rule questions when you are given the following information Angle Side Side in that order (ASS) which should be easy to remember

33 Circle angle theorems A Rule 1 - Any angle in a semi-circle is 90 0 F c B Which angles are equal to 90 0? E D C

34 Big fish?*! Circle angle theorems Rule 2 - Angles in the same segment are equal Which angles are equal here?

35 Circle angle theorems Rule 3 - The angle at the centre is twice the angle at the circumference c c c An arrowhead A little fish A mini quadrilateral c Three radii c Look out for the angle at the centre being part of a isosceles triangle

36 Circle angle theorems Rule 4 - Opposite angles in a cyclic quadrilateral add up to A D C B A + C = and B + D = 180 0

37 A tangent is a line which rests on the outside of the circle and touches it at one point only Circle angle theorems Rule 5 - The angle between the tangent and the radius is 90 0 c

38 Circle angle theorems Rule 6 - The angle between the tangent and chord is equal to any angle in the alternate segment Which angles are equal here?

39 Circle angle theorems Rule 7 - Tangents from an external point are equal (this might create an isosceles triangle or kite) Be prepared to justify these circle theorems by PROVING that c they work

40 Vectors Think of a vector as a journey from one place to another. A vector represents a movement and it has both magnitude (size) and direction H X c Y d L A vector is shown as a line with an arrow on it It can be labelled in two ways: Using a lower case bold letter (usually a or b this is the vector s size) Or using the starting point s letter followed by the destination point s letter with an arrow on top (e.g. GF this shows the direction). Find in terms of c and d, the vectors XY, YX, HL, LH, LY, YL, HX, XH, HY, LX XY = c HL = c LY = d HX = d HY = c + d YX = - c LH = - c YL = - d XH = - d LX = d c

41 Vectors S T P Q R If PS = a, PR = b, Q cuts the line PR in the ratio 2:1 and T cuts the line PS in the ratio 1:3, find the value of : (a) PT (b) SR (c) PQ (d) QT (e) QS (a) PT = ¼ PS so PT = ¼ a (c) PQ = 2/3 PR so PQ = 2/3 b (e) QS =QR + RS so QS = 1/3 b ( a + b) (b) SR Remember = SP + PR SR so = (d) - SR a QT + = b= - a QP so + b+ QS PT = -2/3 so b QT + a= - 2/3 b + ¼a