Topic 1: Pytagoras Teorem Pytagoras Teorem states tat in a rigt anged triange: Te square of te ypotenuse is equa to te sum of te squares of te oter two sides Diagrammaticay: Hypotenuse a c () Oter sides b Te Hypotenuse is te ongest side in te triange and is aso opposite te rigt ange. c = a + b Te square of te Hypotenuse = Sum of te squares of te oter sides Aternativey, because te Hypotenuse is a unique side: = a + b Centre for Teacing and Learning Academic Practice Academic Skis Page 1 T +61 666 96 E ct@scu.edu.au W www.scu.edu.au/teacingearning [ast edited on Juy 015] CRICOS Provider: 0141G
Centre for Teacing and Learning Tis means tat Pytagoras Teorem can be used to find te engt of a missing side in a rigt anged triange. Eampes Hypotenuse unknown (i) 7 9 = a + b rue = 7 + 9 substitute = 49 + 81 = 130 = 130 square root 11.4 (ii) In a rigt triange: side a = 5 cm, b = 10 cm and c is te ypotenuse. Determine te engt of side c. = a + b rue = 5 + 10 substitute = 5 + 100 = 15 = 15 square root 11. Oter side unknown (i) 7 (ii) 9 a = a + b rue = + a 9 7 substitute 841 = 79 + a = a 841 79 rearrange = a 11 square root a = 11 a 10.6 Or = a + b rue b = a rearrange = a s 9 7 ubstutite 841 79 = a = a 11 square root a = 11 a 10.6 5 metre adder 1. metres How far up te wa does te adder reac? = a + b rue 5 = 1. + b substitute 5 = 1.44 + b 5 1.44 = b rearrange 3.56 = b square root b = 3.56 b 4.85 Te adder reaces approimatey 4.85m up te wa. Centre for Teacing and Learning Academic Practice Academic Skis Page T +61 666 96 E ct@scu.edu.au W www.scu.edu.au/teacingearning [ast edited on Juy 015] CRICOS Provider: 0141G
Centre for Teacing and Learning Mied worded questions (i) A sed as a gabe roof as drawn beow. Cacuate te engt of seets of roofing iron required in its construction..4 m 1.6 m 4.8 m Let te engt of te seets be. = a + b = 1.6 +.4 =.56 + 5.76 = 8.3 = 8.3 =.88 Te engt of te seets of roofing iron is.88m (ii) A guy (support) wire is attaced 3. m up a poe and at a point.1 m from te poe. Te ground and te poe are perpendicuar (at rigt anges). Wat is te engt of te guy wire? Poe Guy wire Let te engt of te guy wire be. = a + b = 3. +.1 = 14.65 = 14.65 = 3.83 Te engt of te guy wire is 3.83m Ground (iii) On a softba diamond, te distance between bases is 60 feet or 18.9m. How far must te catcer (at ome base) trow te ba to te payer on second base? Second Base 18.9m 18.9m First Base Te ange at first base is a rigt ange. Let te distance from Home pate to second base be d. = a + b d = 18.9 + 18.9 d = 669.05 d = 669.05 d = 5.87 Te distance from ome pate to second base is 5.87m Home Pate Centre for Teacing and Learning Academic Practice Academic Skis Page 3 T +61 666 96 E ct@scu.edu.au W www.scu.edu.au/teacingearning [ast edited on Juy 015] CRICOS Provider: 0141G
Centre for Teacing and Learning Pytagorean Tripe or Triad Sometimes, te tree engts of a rigt anged triange are a woe numbers. Wen tis occurs, tey are caed a Pytagorean Tripe or Triad. Te most commony known of tese is (3,4,5) representing te triange beow: 3 4 5 = a + b 5 = 3 + 4 5 = 9 + 16 5 = 5 Tree oter Triads are (5,1,13), (7,4,5) and (8,15,17). Oter Triads can be based on mutipes of te base Triads; te Triads (6,8,10), (9,1,15), (1,16,0)... are based te base Triad (3,4,5). If it is important to decide if a triange is a rigt anged triange, ten Pytagoras Teorem can be used to decide tis. 15.4 9.8 10.4 4 19.8 = 19.8 = 39.04 a + b = 15.4 + 9.8 = 37.16 + 96.04 = 333. a + b Tis is not a rigt anged triange. 9.6 = 10.4 = 108.16 a + b = 4 + 9.6 = 16 + 9.16 = 108.16 = a + b Tis is a rigt anged triange. Video Pytagoras Teorem Centre for Teacing and Learning Academic Practice Academic Skis Page 4 T +61 666 96 E ct@scu.edu.au W www.scu.edu.au/teacingearning [ast edited on Juy 015] CRICOS Provider: 0141G
Centre for Teacing and Learning Activity 1. Use Pytagoras Teorem to find te missing engt. (a) (b) 6 18 cm 1.5 cm 0 (c) Find te ypotenuse wen a=6.1 and b=3.4 (d) Find te missing side given a=3.5 and =40 (e) (f) 6 m 6 m 8 m 10 mm Find te eigt of te triange. Aso cacuate te area. (g) A orienteering participant runs 650m nort an ten turns and runs 1.4 km east. How far from te starting point is te runner? () **A Rea Caenge** A square as a diagona of 0 cm, wat is te side engt?. Do te foowing trianges contain a rigt ange? (a) 10, 4, 6 (b) 7, 8, 10 (c), 4.8, 5. (d) 1.4, 4.8, 5 (e) 6, 6, 8 (f) 5, 6, 7 Centre for Teacing and Learning Academic Practice Academic Skis Page 5 T +61 666 96 E ct@scu.edu.au W www.scu.edu.au/teacingearning [ast edited on Juy 015] CRICOS Provider: 0141G