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# Obj. 24 Special Right Triangles

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Identify when a triangle is a 45-45-90 or 30-60-90 triangle
Use special right triangle relationships to solve problems

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### Obj. 24 Special Right Triangles

1. 1. Obj. 24 Special Right Triangles The student is able to (I can): • Identify when a triangle is a 45-45-90 or 30-60-90 triangle • Use special right triangle relationships to solve problems
2. 2. Consider the following triangle: x 1 1 To find x, we would use a2 + b2 = c2, which gives us: 12 + 12 = x 2 x2 = 1 + 1 = 2 x= 2 What would x be if each leg was 2?
3. 3. x 2 2 Again, we will use the Pythagorean Theorem 22 + 22 = x 2 x2 = 4 + 4 = 8 x= 8 Simplifying the radical, we can factor 8 to give us 2 2. Do you notice a pattern?
4. 4. Thm 5-8-1 45º-45º-90º Triangle Theorem In a 45º-45º-90º triangle, both legs are congruent, and the length of the hypotenuse is 2 times the length of the leg. x 45º x 2 45º x
5. 5. Example Find the value of x. Give your answer in simplest radical form. x 1. 8 2 45º 8 2. 7 2 9 2 =9 3. 2 x x 7 9 2
6. 6. If we know the hypotenuse and need to find the leg of a 45-45-90 triangle, we have to divide by 2 . This means we will have to rationalize the denominator, which means to multiply the top and bottom by the radical. 16  16   2  x= =   2  2  2  16 2 = =8 2 2 16 x The shortcut for this is to divide the hypotenuse by 2 and then multiply by 2. 16 x= 2 =8 2 2
7. 7. Examples Find the value of x. 1. x = 20 2 = 10 2 2 45º 20 x 5 2. x = 2 2 x 5
8. 8. Thm 5-8-2 30º-60º-90º Triangle Theorem In a 30º-60º-90º triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg. 30º x 3 2x 60º x Note: the shorter leg is always opposite the 30º angle; the longer leg is always opposite the 60º angle.
9. 9. Examples Find the value of x. Simplify radicals. 1. 14 2. x 11 = 5.5 2 11 x 30º 60º 7 3. 9 3 4. 16 3 =8 3 2 16 16 x x 60º 9 60º
10. 10. To find the shorter leg from the longer leg:  longer leg   3  longer leg 3   = 3  3  3  Examples Find the value of x 1. x = 9 3 =3 3 3 9 60º x 10 3 2. x = 3 10 30º x