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Pythagoras Theorem Reminder of square numbers: 12 = 22 = 32 = 42 = 1x1= 2x2= 3x3= 4x4= 1 4 9 16 Index number 32 Base number The index number tells us how many times the base number is multiplied by itself. e.g. 34 means 3 x 3 x 3 x 3 = 81 1,4,9,16, …. are the answers to a number being squared so they are called square numbers. Pythagoras Theorem means think what is multiplied by itself to make this number? Use your calculator to answer Answer these questions: these questions: 1 1 5.8 2.408 4 2 25.4 5.040 9 3 169 13 16 4 400 20 49 7 1000 31.623 Square root 81 9 8100 90 121 11 225 15 100 10 361 19 Pythagoras Theorem In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. b 2 c c b a 2 Pythagoras of Samos Hypotenuse (6 C BC) a +b 2 a 2 2 =c 2 Pythagoras Theorem To show how this works: b Cut the squares away from the right angle triangle and cut up the segments q of square ‘a’ a Draw line segment x xy, parallel with the hypotenuse of the triangle Draw line segment pq, at right angles to Line segment xy. y p Now rearrange them to look like this. You can see that they make a square with length of side ‘c’. This demonstrates that the areas of squares a and b add up to be the area of square c 2 2 2 a +b =c Pythagoras Theorem x 2 32 4 2 1 3 cm x x 32 42 x 9 + 16 x 25 x 5 cm 4 cm x 2 52 122 2 x 5 cm x 5 12 2 x 12 cm 169 x 13 cm 2 Pythagoras Theorem 3 5 cm x 2 52 6 2 x x 52 62 x 7.8 cm (1 dp) 6 cm x 2 4.62 9.82 4 x 4.6 cm 9.8 cm x 4.6 9.8 x 10.8 cm (1 dp) 2 2 x 2 252 72 Pythagoras Theorem x 252 72 x 11 9 2 Now do these: 5 11m 2 x 11 9 2 2 xx m24 m 8 x 6.3 m (1 dp) xm 9m 6 11 cm 2 x 23.8 11 2 23.8 cm 3.4 cm 7.1 cm x cm 2 2 x 23.82 112 x 21.1 cm (1 dp) xm 7 25 m x 2 7.12 3.4 2 x 7.12 3.42 x 7.9 cm (1 dp) 7m Pythagoras Theorem A boat sails due East from a Harbour (H), to a marker buoy (B),15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. What is the total distance travelled by the boat? H 15 miles B LH 2 152 6.42 6.4 miles LH 152 6.42 LH 16.3 miles Total distance travelled = 21.4 + 16.4 = 37.7 miles L Pythagoras Theorem A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. How far is the base of the ladder from the house? L2 122 9.52 L 122 9.52 12 ft 9.5 ft L 7.3ft L