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**Two Special Right Triangles**

45°- 45°- 90° 30°- 60°- 90°

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**The 45-45-90 triangle is based on the square with sides of 1 unit.**

45°- 45°- 90° The triangle is based on the square with sides of 1 unit. 1

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**If we draw the diagonals we form two 45-45-90 triangles.**

45°- 45°- 90° If we draw the diagonals we form two triangles. 1 45° 45° 45° 45°

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**Using the Pythagorean Theorem we can find the length of the diagonal.**

45°- 45°- 90° Using the Pythagorean Theorem we can find the length of the diagonal. 1 45° 45° 45° 45°

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**12 + 12 = c2 1 + 1 = c2 2 = c2 2 = c 45°- 45°- 90° 45° 45° 2 1 45°**

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**Conclusion: the ratio of the sides in a 45-45-90 triangle is 1-1-2**

45°- 45°- 90° Conclusion: the ratio of the sides in a triangle is 1-1-2 1 2 45°

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45°- 45°- 90° Practice 4 45° 4 2 4 SAME leg*2

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45°- 45°- 90° Practice 9 45° 9 2 9 SAME leg*2

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45°- 45°- 90° Practice 2 45° 2 2 2 SAME leg*2

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45°- 45°- 90° Practice 7 45° 14 7 SAME leg*2

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45°- 45°- 90° Practice Now Let's Go Backward

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45°- 45°- 90° Practice 45° 3 2 hypotenuse2

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45°- 45°- 90° Practice 3 2 2 = 3

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45°- 45°- 90° Practice 45° 3 2 3 3 SAME hypotenuse2

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45°- 45°- 90° Practice 45° 6 2 hypotenuse2

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45°- 45°- 90° Practice 6 2 2 = 6

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45°- 45°- 90° Practice 45° 6 2 6 6 SAME hypotenuse2

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45°- 45°- 90° Practice 45° 11 2 hypotenuse2

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45°- 45°- 90° Practice 11 2 2 = 11

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45°- 45°- 90° Practice 45° 112 11 11 SAME hypotenuse2

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45°- 45°- 90° Practice 45° 8 hypotenuse2

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45°- 45°- 90° Practice 8 2 2 * = 82 2 = 42

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45°- 45°- 90° Practice 45° 8 42 42 SAME hypotenuse2

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45°- 45°- 90° Practice 45° 4 hypotenuse2

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45°- 45°- 90° Practice 4 2 2 * = 42 2 = 22

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45°- 45°- 90° Practice 45° 4 22 22 SAME hypotenuse2

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45°- 45°- 90° Practice 45° 6 Hypotenuse 2

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45°- 45°- 90° Practice 6 2 2 * = 62 2 = 32

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45°- 45°- 90° Practice 45° 6 32 32 SAME hypotenuse2

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30°- 60°- 90° The triangle is based on an equilateral triangle with sides of 2 units. 2 60°

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30°- 60°- 90° The altitude (also the angle bisector and median) cuts the triangle into two congruent triangles. 2 60° 30° 30° 1 1

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30°- 60°- 90° This creates the triangle with a hypotenuse a short leg and a long leg. 30° 60° Long Leg hypotenuse Short Leg

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30°- 60°- 90° Practice We saw that the hypotenuse is twice the short leg. 60° 30° 2 We can use the Pythagorean Theorem to find the long leg. 1

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**2 1 3 30°- 60°- 90° Practice A2 + B2 = C2 A2 + 12 = 22 A2 + 1 = 4**

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**Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- 2 - 3**

30°- 60°- 90° 60° 30° Conclusion: the ratio of the sides in a triangle is 3 2 3 1

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30°- 60°- 90° Practice 60° 30° The key is to find the length of the short side. 8 43 Hypotenuse = short leg * 2 4 Long Leg = short leg * 3

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**10 53 5 30°- 60°- 90° Practice Hypotenuse = short leg * 2**

Long Leg = short leg * 3

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**14 73 7 30°- 60°- 90° Practice Hypotenuse = short leg * 2**

Long Leg = short leg * 3

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**23 3 3 30°- 60°- 90° Practice Hypotenuse = short leg * 2**

Long Leg = short leg * 3

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**210 30 10 30°- 60°- 90° Practice Hypotenuse = short leg * 2**

Long Leg = short leg * 3

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30°- 60°- 90° Practice Now Let's Go Backward

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**22 113 11 30°- 60°- 90° Practice Short Leg = Hypotenuse 2**

Long Leg = short leg * 3

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**4 23 2 30°- 60°- 90° Practice Short Leg = Hypotenuse 2**

Long Leg = short leg * 3

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**18 93 9 30°- 60°- 90° Practice Short Leg = Hypotenuse 2**

Long Leg = short leg * 3

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**30 153 15 30°- 60°- 90° Practice Short Leg = Hypotenuse 2**

Long Leg = short leg * 3

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**46 233 23 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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**28 143 14 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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**32 163 16 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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**9 3 3 63 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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**12 4 3 83 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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**27 9 3 183 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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**21 7 3 143 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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**33 113 223 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2**

Short Leg = Long leg 3

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THE END

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