# two special right triangles 45°- 45°- 90° 30°- 60°- 90°

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Two Special Right Triangles 45- 45- 90 30- 60- 90 Slide 2 45- 45- 90 The 45-45-90 triangle is based on the square with sides of 1 unit. The 45-45-90 triangle is based on the square with sides of 1 unit. 1 1 1 1 Slide 3 45- 45- 90 If we draw the diagonals we form two 45-45-90 triangles. If we draw the diagonals we form two 45-45-90 triangles. 1 1 1 1 45 Slide 4 45- 45- 90 Using the Pythagorean Theorem we can find the length of the diagonal. Using the Pythagorean Theorem we can find the length of the diagonal. 1 1 1 1 45 Slide 5 45- 45- 90 1 2 + 1 2 = c 2 1 + 1 = c 2 2 = c 2 2 = c 1 2 + 1 2 = c 2 1 + 1 = c 2 2 = c 2 2 = c 1 1 1 1 45 22 Slide 6 45- 45- 90 Conclusion: the ratio of the sides in a 45-45-90 triangle is 1-1- 2 Conclusion: the ratio of the sides in a 45-45-90 triangle is 1-1- 2 1 1 22 45 Slide 7 45- 45- 90 Practice 4 4 2 SAME leg* 2 4 45 Slide 8 45- 45- 90 Practice 9 9 2 SAME leg* 2 9 45 Slide 9 45- 45- 90 Practice 2 2 2 SAME leg* 2 2 45 Slide 10 45- 45- 90 Practice 14 SAME leg* 2 77 77 45 Slide 11 45- 45- 90 Practice Slide 12 3 2 hypotenuse 2 45 Slide 13 45- 45- 90 Practice 3 2 22 = 3 Slide 14 45- 45- 90 Practice 3 2 hypotenuse 2 45 3 SAME 3 Slide 15 45- 45- 90 Practice 6 2 hypotenuse 2 45 Slide 16 45- 45- 90 Practice 6 2 22 = 6 Slide 17 45- 45- 90 Practice 6 2 hypotenuse 2 45 6 SAME 6 Slide 18 45- 45- 90 Practice 11 2 hypotenuse 2 45 Slide 19 45- 45- 90 Practice 11 2 22 = 11 Slide 20 45- 45- 90 Practice 11 2 hypotenuse 2 45 11 SAME 11 Slide 21 45- 45- 90 Practice 8 hypotenuse 2 45 Slide 22 45- 45- 90 Practice 8 22 22 22 * = 8282 2 = 4 2 Slide 23 45- 45- 90 Practice 8 hypotenuse 2 45 4242 SAME 4242 Slide 24 45- 45- 90 Practice 4 hypotenuse 2 45 Slide 25 45- 45- 90 Practice 4 22 22 22 * = 4242 2 = 2 2 Slide 26 45- 45- 90 Practice 4 hypotenuse 2 45 2222 SAME 2222 Slide 27 45- 45- 90 Practice 6 Hypotenuse 2 45 Slide 28 45- 45- 90 Practice 6 22 22 22 * = 6262 2 = 3 2 Slide 29 45- 45- 90 Practice 6 hypotenuse 2 45 3232 SAME 3232 Slide 30 30- 60- 90 The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units. The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units. 2 2 2 60 Slide 31 2 2 2 30- 60- 90 The altitude (also the angle bisector and median) cuts the triangle into two congruent triangles. The altitude (also the angle bisector and median) cuts the triangle into two congruent triangles. 1 1 30 Slide 32 60 This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg. This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg. 30- 60- 90 hypotenuse Short Leg Long Leg Slide 33 60 30 30- 60- 90 Practice 1 2 We saw that the hypotenuse is twice the short leg. We saw that the hypotenuse is twice the short leg. We can use the Pythagorean Theorem to find the long leg. We can use the Pythagorean Theorem to find the long leg. Slide 34 60 30 30- 60- 90 Practice 1 2 33 A 2 + B 2 = C 2 A 2 + 1 2 = 2 2 A 2 + 1 = 4 A 2 = 3 A = 3 A 2 + B 2 = C 2 A 2 + 1 2 = 2 2 A 2 + 1 = 4 A 2 = 3 A = 3 Slide 35 30- 60- 90 Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- 2 - 3 Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- 2 - 33 60 30 33 1 2 Slide 36 60 30 30- 60- 90 Practice 4 8 Hypotenuse = short leg * 2 4343 The key is to find the length of the short side. The key is to find the length of the short side. Long Leg = short leg * 3 Slide 37 60 30 30- 60- 90 Practice 5 10 Hypotenuse = short leg * 2 5353 Long Leg = short leg * 3 Slide 38 60 30 30- 60- 90 Practice 7 14 Hypotenuse = short leg * 2 7373 Long Leg = short leg * 3 Slide 39 60 30 30- 60- 90 Practice 33 2323 Hypotenuse = short leg * 2 3 Long Leg = short leg * 3 Slide 40 60 30 30- 60- 90 Practice 10 2 10 Hypotenuse = short leg * 2 30 Long Leg = short leg * 3 Slide 41 30- 60- 90 Practice Slide 42 60 30 30- 60- 90 Practice 11 22 Short Leg = Hypotenuse 2 11 3 Long Leg = short leg * 3 Slide 43 60 30 30- 60- 90 Practice 2 4 Short Leg = Hypotenuse 2 2323 Long Leg = short leg * 3 Slide 44 60 30 30- 60- 90 Practice 9 18 Short Leg = Hypotenuse 2 9393 Long Leg = short leg * 3 Slide 45 60 30 30- 60- 90 Practice 15 30 Short Leg = Hypotenuse 2 15 3 Long Leg = short leg * 3 Slide 46 60 30 30- 60- 90 Practice 23 46 Hypotenuse = Short Leg * 2 23 3 Short Leg = Long leg 3 Slide 47 60 30 30- 60- 90 Practice 14 28 Hypotenuse = Short Leg * 2 14 3 Short Leg = Long leg 3 Slide 48 60 30 30- 60- 90 Practice 16 32 Hypotenuse = Short Leg * 2 16 3 Short Leg = Long leg 3 Slide 49 60 30 30- 60- 90 Practice 3 3 6363 Hypotenuse = Short Leg * 2 9 Short Leg = Long leg 3 Slide 50 60 30 30- 60- 90 Practice 4 3 8383 Hypotenuse = Short Leg * 2 12 Short Leg = Long leg 3 Slide 51 60 30 30- 60- 90 Practice 9 3 18 3 Hypotenuse = Short Leg * 2 27 Short Leg = Long leg 3 Slide 52 60 30 30- 60- 90 Practice 7 3 14 3 Hypotenuse = Short Leg * 2 21 Short Leg = Long leg 3 Slide 53 60 30 30- 60- 90 Practice 11 3 22 3 Hypotenuse = Short Leg * 2 33 Short Leg = Long leg 3 Slide 54