Example 2. If the area of this square is 144 cm 2, find the length of one of its sides.

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1 THEOREM OF PYTHAGORAS Introduction: Squares and Square Roots Students shold be shon both the 2 and Ö button on calculator. This eercise is simply based on revision of: (i) the use of a calculator to square and square root. (ii) the use of A = L 2, the formula for finding the area of a square. The folloing eamples can be used: Eample 1. Find the area of this square: Ans. A = L 2 = 5 2 = 25 cm 2 5 cm Eample 2. If the area of this square is 144 cm 2, find the length of one of its sides. Ans. A = L = L 2 Ö144 = L L = 12 cm Area = 144 cm 2 Eercise 1 may no be attempted A. The Theorem of Pythagoras Finding the length of one side of a right angled triangle, given the length of the other to sides. (a) Finding the longest side: The term Hypotenuse should be eplained. Then the folloing eample can be given: 5 3 Ask the students to ork out the area of each of the three squares. Ans. 9, 16, Ask them to do the same again ith this shape. Ans. 36, 64, Perhaps do another, ith squares of sides 5, 12 and Mathematics Support Materials: Mathematics 2 (Int 1) Staff Notes 8

2 From these ansers it can be deduced that: the area of the square on the shortest side the area of the square on the longest side = + the area of the square on the middle side In summary, in a right angled triangle ith sides a, b and c. a 2 = b 2 + c 2 c a b When calculating the longest side (hypotenuse) in a right angled triangle, use Pythagoras Plus. Eample 1. Write an equation for. Ans. 2 = Eample 2. Calculate, the longest side (correct to 1 decimal place if necessary). Ans. 2 = (correct formula + ) = (square out) = 100 (tidy) = Ö100 (bring in Ö) = 10 (use calc.) 8 Eample 3. Calculate, (correct to 1 decimal place if necessary). 6 Ans. 2 = (correct formula + ) = (square out) = 250 (tidy) = Ö250 (bring in Ö) = 15 8 (use calc. and round) 13 9 Eercise 2 may no be attempted Mathematics Support Materials: Mathematics 2 (Int 1) Staff Notes 9

3 (b) Finding one of the shorter sides: The difference beteen the hypotenuse and the shorter sides should be shon. For this case, change the original formula to arrive at: b 2 = a 2 c 2 Of the 2 sides given, this number should alays be the larger one. c a or c 2 = a 2 b 2 b When calculating one of the shorter sides (not the hypotenuse) in a right angled triangle - use Pythagoras Minus. Eample 1. Write an equation for : Ans. 2 = the larger of the to numbers given. Eample 2. Calculate the missing side, (correct to 1 decimal place if necessary). Ans. 2 = (correct formula - ) = (square out) = 64 (tidy) = Ö64 (bring in Ö) = 8 (use calc.) 6 10 Eample 3. Calculate, correct to 1 decimal place. Ans. 2 = (correct formula - ) = (square out) = (tidy) = Ö31 71 (bring in Ö) = 5 6 (m) (use calc. and round) 8 6 m 6 5 m Mathematics Support Materials: Mathematics 2 (Int 1) Staff Notes 10 m

4 Eercise 3 may no be attempted (c) A miture of Pythagoras Plus and Pythagoras Minus It should be eplained to students that they must first make the folloing decision - if the hypotenuse is asked for... use the + formula if one of the shorter sides (not the hypotenuse) is asked for... use the formula. The folloing eamples can be used: Eample 1. What length of cable is needed to secure this flag pole? Ans. Hypotenuse asked for => Pythagoras Plus 2 = (correct formula + ) = (square out) = (tidy) = Ö97 25 (bring in Ö) = 9 9 (m) (use calc. and round) 8 5m 5m Eample 2. A 6m pipe is resting against the top of a all.the other end of the pipe is sitting on the ground, 3m from the foot of the all. What is the height of the all? Ans. Shorter side asked for, (not hypotenuse) => Pythagoras Minus 2 = (correct formula - ) = 36 9 (square out) = 27 (tidy) = Ö27 (bring in Ö) = 5 2 (m) (use calc. and round) 6m Eercise 4 Questions 1-12 may no be attempted (Q13 to be used as an etension). 3m Mathematics Support Materials: Mathematics 2 (Int 1) Staff Notes 11

5 (d) Finding the distance beteen to coordinate points. It may be that some revision on coordinates ill be required first. The folloing eample can be used: Eample Ans. Find the distance PQ, beteen the points P( 2,3) and Q(7,6). y Q Plot the points correctly on a diagram or sketch. P y Q Make a right angled triangle by draing a vertical line through Q and a horizontal one through P. P 9 boes along 3 boes up Proceed as if you are finding the hypotenuse of a right angled triangle => Pythagoras Plus P 9 Q 3 2 = (correct formula + ) = (square out) = 90 (tidy) = Ö90 (bring in Ö) = 9 5 (use calc. and round) PQ has length 9 5 units Eercise 5 Q1, Q2 and Q3 may no be attempted. Q4 for etension (converse) Then do the checkup for Theorem of Pythagoras. Mathematics Support Materials: Mathematics 2 (Int 1) Staff Notes 12

6 Squares and Square Roots Eercise 1 1. Find the value of: (a) 5 2 (b) 8 2 (c) 10 2 (d) 1 2 (e) (f) (g) (h) Find the value of: (a) Ö25 (b) Ö81 (c) Ö100 (d) Ö20 25 (e) Ö (f) Ö324 (g) Ö (h) Ö1 3. Calculate the area of these squares, giving your ansers correct to 1 decimal place: (a) (b) (c) 3 2cm 5 5cm 6 9cm 4. Calculate the length of a side in each of these squares: (a) (b) (c) area = 9cm 2 area = 56 25cm 2 area = 29 16cm 2 The Theorem of Pythagoras Eercise 2 1. Use Pythagoras Theorem to rite an equation for each of these triangles: (the first one has been done for you) (a) r q r 2 = p 2 + q 2 p * r is the longest side. contd... Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 21

7 (b) (c) (d) z p m c b y t m a s m 2. Calculate the length of the missing side. Give your ansers correct to one decimal place. (a) (b) (c) (d) (e) 5 60 (f) (g) 12 2 (h) 12mm mm (i) 6 5cm 16mm cm 8 5cm (j) (k) m 1 5 m m 5 m 3 2 m 7 6 m Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 22

8 Eercise 3 1. Use Pythagoras Theorem to rite an equation hich can be used to calculate the required side in each of the folloing triangles: (the first one has been done for you) (a) v u 2 = v 2 2 u here u is one of the to shorter sides. (b) Write an equation for finding a here. c b (c) Write an equation for finding b here. (d) Write an equation for finding y here. b a m t f m y m b m 2. Calculate the length of the missing side, giving your ansers correct to one decimal place. 6 (a) (b) (c) (d) (e) (f) contd... Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 23

9 (g) 800 (h) mm 2mm mm (i) (j) 2 84 m 1 8 m 9 6 m m m 7 5 m Eercise 4 In this eercise, give all your ansers correct to 1 decimal place. 1. Calculate the height of this set square cm h cm 9 cm 2. A 1 5 metre ooden post is cemented vertically into the ground and requires support until the cement dries. Due to marshy conditions, the nearest spot here a peg can be hammered in to hold a supporting ire is 2 1 metres from the post. What is the minimum length of ire hich ill be required? ire 2 1m 1 5m 3. A fighter pilot flies 230 kilometres 110km due East from base. He then flies 110 kilometres due North. Ho far is he no from base? Base 230km 4. A gate hich is 3 3 metres ide has a 4 metre ooden diagonal support. Calculate the height of the gate? 4m 3 3m h m Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 24

10 5. In order to rescue a cat ho finds herself stuck in a gutter at the top of a 4 8 metre brick all, the rescuer places his 5 metre ladder against the side of the all. For safety, he has to place the foot of the ladder at least 1 metre from the base of the all. Find ho far the base of his ladder is from the the all and state if it is safe for him to complete the rescue. 5 m 4 8 m 6. Ted and his father are flying a kite. 60 m When Ted has let out 60 metres of string the kite is 40 metres above the 40 m ground and directly above his father. Ho far aay from his father is Ted standing? Ted Father 7. Calculate the height of this tree hich is supported by a 10 metre rope tied don 8 5 metres from the foot of the tree. 10m h m 8 5m 8. What length of cable is needed to secure this flag pole? 7 5m 6m 9. The drabridge of a castle is supported by a 3 4m 3 4 metre chain hich is attached to a bolt 8 5 metres up the castle all. 8 5m Calculate the length of the drabridge. Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 25

11 10. The picture shos an end vie of a house etension. The top part is made of timber. 3 8 m 2 1 m H 3 1 m (a) Calculate the height ( metres) of the timber part of the etension. (b) What is the overall height (H metres) of the etension? B 11. This picture shos a telegraph pole ith 2 ires connecting the top of the pole to the ground. 8m (a) Calculate the height of the telegraph pole. (b) Use the anser to part (a) to ork out the A 5m 9m C length of ire BC. D Q R 12. PSTU is a rectangular face of this cuboid. PU = 5mm and PT = 13mm. Calculate the length of the line UT. 5mm P 13mm S W U? T 13. HARD! The baby portrait is centimetres in length. (Line AB = centimetres) Calculate the length of AC, half of the string used for hanging the picture. C C A B A cm B cm Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 26

12 Eercise 5 1. Calculate the lengths of the 5 lines, AB, CD, EF, GH and IJ giving your anser correct to 1 decimal place. y D C B E 2 A 1 G J -4 H -5 F -6-7 I Plot these points on a coordinate diagram and calculate the lengths of the lines joining them. (a) P(1,2) and Q(9,8) (b) R(2, 1) and S( 3,11) 3. Part of a tiled bathroom all is shon. A piece of sloping ceiling cuts across the tiles. The square tiles measure 30 centimetres by 30 centimetres. Calculate the length of the sloping ceiling from A to B. 30cm 30cm A B 4. Charles as asked to dra a triangle ith sides 32 8 centimetres, 24 6 centimetres and 41 centimetres. He dre a triangle ith the correct measurements 32 8cm but sketched it like the one shon belo. Eplain hy Charles triangle should really have had a right angle in it. 41cm 24 6cm Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 27

13 Mathematics 2 (Intermediate 1) Checkup for The Theorem of Pythagoras 1. Calculate the length of the unknon side, correct to 1 decimal place, in each of these triangles: (a) 10 cm cm (b) 9 cm 4 cm 24 cm cm 17 cm (c) 4 5 cm 6 cm (d) 24 cm h cm p cm 1 3mm (e) (f) 12cm 12 8cm 7 8mm s mm y cm 2. A bus is sitting on a giant ramp. Find the length of the ramp.? 12 m 24 m 3. h cm 86 cm This diagram shos the sail on a model yacht. Calculate the height of the sail. 70 cm Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 28

14 4. A church steeple is in the shape of an isosceles triangle. It is 15 metres high and has a sloping edge of 15 7 metres. Calculate: (a) the length L (metres). (b) the idth W (metres) of the steeple. 15m 15 7m W L 5. Plot these points on a coordinate diagram and calculate the lengths of the lines joining them. (a) A(1,0) and B(7,8) (b) C( 4, 2) and D( 7, 6) Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 29

15 ANSWERS TO MATHEMATICS 2 (INT 1) Theorem of Pythagoras Eercise 1 1. (a) 25 (b) 64 (c) 100 (d) 1 (e) (f) 0 09 (g) (h) (a) 5 (b) 9 (c) 10 (d) 4 5 (e) 10 5 (f) 18 (g) 100 (h) 1 3. (a) 10 2cm 2 (b) 30 3cm 2 (c) 47 6cm 2 4. (a) 3cm (b) 7 5cm (c) 5 4cm Eercise 2 1. (a) done (b) c 2 = a 2 + b 2 (c) 2 = y 2 + z 2 (d) p 2 = t 2 + s 2 2. (a) 13( 0) (b) 29( 0) (c) 5( 0) (d) 15( 0) (e) 100( 0) (f) 7 6 (g) 12 2 (h) 20( 0) (i) 10 7 (j) 3 5 (k) 9 1 Eercise 3 1. (a) done (b) a 2 = c 2 - b 2 (c) b 2 = m 2 - t 2 (d) y 2 = f 2 - b 2 2. (a) 5( 0) (b) 6 7 (c) 46 6 (d) 94 7 (e) 7( 0) (f) 71 4 (g) 600( 0) (h) 1 2 (i) 2 2 (j) 6( 0) Eercise cm m ( 0)km m m, safe to complete rescue m m m m 10. (a) 2 2m (b) 4 3m 11. (a) 6 2m (b) 11( 0)m mm ( 0)cm Eercise 5 1. AB = 5 4, CD = 10 4, EF = 7 1, GH = 8 1, IJ = (a) 10 (b) cm = 41 2 Check Up 1. (a) 26( 0) cm (b) 8 1cm (c) 7 5cm (d) 16 9 cm (e) 4 5cm (f) 7 9mm m 3. 50( 0)cm 4. (a) 4 6m (b) 9 2/9 3m 5. (a) 10 (b) 5 Mathematics Support Materials: Mathematics 2 (Int 1) Student Materials 36