Applying Pythagorean Theorem

Transcription

1 Applying Pythagorean Theorem Pythagoras was born in the late 6th entury BC on the island of Samos. He was a Greek philosopher and religious leader who was responsible for important developments in mathematis, astronomy and the theory of musi. Pythagoras is famous also for the fat that he allegedly paid for his first student. Frustrated that no one would listen to his learning, he deided to "buy" a student. It is believed that the Egyptians and other anient ultures used a rule in onstrution. In Egypt, Pythagoras studied with the engineers, known as "rope strethers", who built the pyramids. They had a rope with 12 evenly spaed knots. If the rope was pegged to the ground in the dimensions 3 4 5, a right triangle would result. This enabled them to lay the foundations of their buildings aurately. 1

2 Right Triangles a b b a b a All right angle triangles are labelled the same way the side opposite the right angle is alled the hypotenus It is always labelled "" the other two sides are labelled "b" for base and "a" for altitude 2

3 In any right angle triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides. a 2 + b 2 = 2 Some number groups are known as Pythagorean triplets or triads. They an be used to make up the sides of a right angle triangle ie: 3, 4, 5 a 2 + b 2 = = = = 25 ie: 2, 6, 10 a 2 + b 2 = 2 2, 6, 10 is not a triplet = = This would not be a right angle triangle 3

4 We an use the Pythagorean Theorem with area 16m 2 25m 2 9m 2 The area of a square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides. Whole Area of a + Whole Area of b = Whole Area of 4

5 To find the side length of a right angle triangle square root of the area. you find the b a Area = 25m 2 The length of side b = 25 = 5 b = 5 5

6 Finding Missing Measurements of side lengths using the Pythagorean Theorem a) 12m 16m a 2 + b 2 = = = = = 20 = b) 3m a 2 + b 2 = = = 2 25 = 2 25 = 5 = 4m Using Pythagorean Theorem find the hypotenuse... i) a = 7m, b = 6m ii) a = 2.5m, b = 4.1m 6

7 If you have to find a or b, then you go bakwards and do the opposite you subtrat 2 b 2 = a 2 or 2 a 2 = b 2 Example 1: b 10m 6m 2 a 2 = b = b = b 2 64 = b 2 64 = b 8 = b Example 2: a 25m 2 b 2 = a = a = a 2 49 = a 2 49 = a 7 = a 24m Try these... i) a = 5m, = 12m, find b ii) b = 6.8m, = 9.7m, find a Book Work... Page 34 #'s 3, 4, 5, 6, 7, 8, 9, 13 Worksheets Book Work... Page 43 #'s 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 7

8 Remember Pythagorean Theorem a 2 + b 2 = 2 This an be used to solve story problems that involve right angle triangles Example 1: The size of a tv sreen is the length of its diagonal. A tv sreen is 56m wide and 40m high. Calulate the length of its diagonal to the nearest entimeter. b = 56m a = 40m a 2 + b 2 = = = = = 69 = 8

9 Example #2: Anna & Liam are on a hike. They ome to a orner of a retangular field that measures 750m by 400m. Anna deides to take a shortut and walk diagonally aross the field. Liam walks around two sides of the field. a) Who walks farther? b) How muh farther does that person walk? 750m 400m Anna & Liam Liam: = 1150m Anna: a 2 + b 2 = = = = = 850 = a) Liam walked farther, he walked 1150m, whereas, Anna walked 850m b) = 300 Liam walked 300 m farther 9

10 Example #3: In the RST, alulate the length of RS to the nearest millimeter 6m?? 10m 2 a 2 = b = b = b 2 64 = b 2 64 = b 8 = b 8m = 80mm Worksheets 19 & 20 Book work: page 49 #'s 6, 7, 8, 9, 10, 11, 12, 13, 16, 17 10