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Special Right Triangles: Properties and Examples of 30-60-90 and 45-45-90 Triangles

This lesson focuses on special right triangles, specifically the 30-60-90 and 45-45-90 triangles. In a 45-45-90 triangle, both legs are congruent, and the hypotenuse is √2 times the length of a leg. Conversely, in a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times that. Through examples and step-by-step instructions, we will determine the lengths of the sides of these triangles, enhancing your understanding of their properties and applications.

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Special Right Triangles: Properties and Examples of 30-60-90 and 45-45-90 Triangles

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  1. Unit 7 Part 2 Special Right Triangles 30°, 60,° 90° ∆s 45°, 45,° 90° ∆s

  2. Special Right Triangles • In a 45-45-90 degrees right triangle both legs are congruent and the hypotenuse is the length of the leg times 2 45 1 45 1

  3. Another way of stating the formula 45-45-90 Triangle In a 45-45-90 triangle, the length of the hypotenuse is 2 times the length of one leg. x x x

  4. Example • Determine the length of each side of the following 45-45-90 triangle. w 45 n 1 45 5 1

  5. Find the length of each side. 45 8 √2 8 2 n 45 w 45 1 45 1

  6. Find the length of each side. • The hypotenuse is 6 2 45 6 2 n 45 w 45 1 45 1

  7. Find the length of each Variable • This is a 45-45-90 triangle. y x 3 45 1 45 1

  8. 30-60-90 triangle • In a 30-60-90 right triangle this is the format. 2 60 1 30 x 2x 30 60 1 x Hypotenuse = 2 Adjacent to 30 = Adjacent to 60 = 1 60 2 30

  9. Another way of stating the formula 30-60-90 Triangles In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg. 2x 60 x 30 x 3

  10. Example • Determine the length of each side of the following 30-60-90 triangle. 2 60 16 n 1 30 30 w

  11. Find the length of each variable. 2 60 1 30 x 60 y 30 5 √ 3 5  3

  12. Find the length of each side. 30 30 2 12 n 60 1 60 w

  13. Find the length of each variable. 2 60 1 30 s 60 r 30 10

  14. 30 45 45 60

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