Special Right Triangles 30:60:90 Right Triangles

30:60:90 Relationship Given: Equilateral Triangle with side=2, find the altitude. 30º 2 2 x 60º 1 2

30:60:90 Relationship Given: Equilateral Triangle with side=4, find the altitude. 30º 4 4 x 60º 2 4

30:60:90 Relationship Given: Equilateral Triangle with side=10, find the altitude. 30º 10 10 x 60º 5 10

Conclusion - The side opposite the 30º angle is half the hypotenuse - The side opposite the 60º angle is half the hypotenuse times 2 1 - The ratio of the sides of a 30:60:90 right triangle is 60º 1

Since 30 is the smallest angle, then the 1 goes across from it! Page 25 Remember, the 30-60-90 triangle always has the same ratio for its sides: Remember the relationship the sides have with the angles! The smallest side is across from the smallest angle! Since 30 is the smallest angle, then the 1 goes across from it! Since 60 is the next biggest angle, then the 3 goes across from it! Since 90 is the largest angle, then the 2 goes across from it!

Page 26 30:60:90 Now, since the ratio is always the same, then what did we multiply by? 1: 3 :2 5: 5 3 :10 Five! If we multiply one number in the ratio by 5, we multiply all of them by 5. 𝑦=5 3 𝑥=10

Page 26 30:60:90 1: 3 :2 Multiply by 10 10:10 3 :20 𝑦=10 3 𝑥=10

Page 26 30:60:90 1: 3 :2 Multiply by 1 1 2 : 3 : 𝑥=1 𝑦=2

Page 26 30:60:90 1: 3 :2 Multiply by 15 15:15 3 :30 𝑥=15 𝑦=15 3

Page 26 30:60:90 2∙ 30 3 = 60 3 𝑥∙ 3 =30 1: 3 :2 𝑥∙ 3 3 = 30 3 30 3 60 3 :30: 𝑥= 30 3 𝑥= 30 3 𝑦= 60 3 If you see this on the Regents and is a multiple choice question, compare decimals to the answer given. 𝑥= 30 3 =17.32050808

Page 26 30:60:90 1: 3 :2 Multiply by 5 30 5: :10 5 3 10 10 𝑥 60 90 5 60 10

Page 26 30:60:90 1: 3 :2 Multiply by 8 30 8: :16 8 3 16 16 𝑥 60 90 8 60 16

Page 26 30:60:90 1: 3 :2 Multiply by 2 30 2: :4 2 3 4 4 𝑥 60 90 2 60 4

Page 26 30:60:90 1: 3 :2 Multiply by 3.5 30 3.5: :7 3.5 3 7 7 𝑥 60 90 3.5 60 7

Page 26 30:60:90 1: 3 :2 Multiply by 7.5 30 7.5: :15 7.5 3 15 15 𝑥 60 90 7.5 60 15

Page 26 30:60:90 1: 3 :2 Multiply by 𝑠 2 30 𝑠 2 : :𝑠 𝑠 2 3 𝑠 𝑠 𝑥 𝑠 2 60 90 60 𝑠

Page 26 30:60:90 1: 3 :2 Multiply by 7 30 7 14 :7 3 : 14 7 3 60 90 7 60

Page 26 30:60:90 1: 3 :2 Multiply by 5 60 10 5:5 3 :10 5 30 90 14

Page 26 30:60:90 1: 3 :2 Multiply by 4 60 4:4 3 :8 8 90 30 4 3 4 3 =6.92=6.9

Page 26 60 10 60 5 5 5 3 10 30 90 90 5 3 5 3 5 30 60 30 30:60:90 1: 3 :2 𝑆ℎ𝑜𝑟𝑡𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑖𝑠 5+5=10 Multiply by 5 5:5 3 :10 𝐿𝑜𝑛𝑔𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑖𝑠 5 3 +5 3 =10 3

Page 26 8 3 8 4 3 12 𝑥∙2=8 3 4 𝑥∙ 2 2 = 8 3 2 30:60:90 30:60:90 𝑥=4 3 Multiply by 4 1: 3 :2 1: 3 :2 4 3 : 12 :8 3 4 8 :4 3 : 3 ∙4 3 =4 9 =4∙3 =12 𝐴𝐵=4+12=16

Homework Page 26 #12,14,16,19 Separate Sheet

Page 26 A B C 30:60:90 60 120 1: 3 :2 8 Multiply by 4 4:4 3 :8 4 30 90 4 3 4 3 4 3 +4 3 =13.8 =14

180−120=60 Page 26 60 2 =30 30 30 10 3 10 5 3 30 120 60 10 30:60:90 15 1: 3 :2 Multiply by 5 30:60:90 5: 5 3 :10 1: 3 :2 Multiply by 5 3 5 3 : 15: 10 3

Page 26 60 6 3 30 6 3 3 30:60:90 3 1: 3 :2 Multiply by 3 3:3 3 :6 12 30:60:90 1: 3 :2 Multiply by 6 6: 6 3 :12

Page 26 30:60:90 12 3 1: 3 :2 12 6 3 Multiply by 6 :6 3 : 6 12 6 24−6=18 30:60:90 1: 3 :2 Multiply by 12 24 12: 12 3 :24