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Understanding Special Right Triangles and the Pythagorean Theorem

This guide explores calculating the hypotenuse and sides of special right triangles, specifically 30-60-90 and 45-45-90 triangles. It covers how to solve equations for unknown sides and provides step-by-step examples, including using the Pythagorean theorem and trigonometric ratios. Learn the relationships between the sides, such as the hypotenuse being twice the length of the shorter leg and finding values in simplest radical form. Perfect for students seeking to understand these key geometric principles.

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Understanding Special Right Triangles and the Pythagorean Theorem

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  1. Warm Up April 14th • What is the hypotenuse if the leg lengths are a = 6 and b = 4? • Simplify

  2. EOCT Week 13 #1

  3. Solve for x 45 x 3 3

  4. Solve for x 45 3√2 x X=3

  5. Solve for x 45 3 x 3=x√2

  6. Special Right Triangles You will be able to find the lengths of sides of special right triangles 30-60-90 And 45-45-90

  7. 45-45-90 Right Triangle x x

  8. 60 2a a 30 30-60-90 Right Triangle

  9. 30-60-90 Triangles The longer leg is _____ times bigger than the shorter leg. The hypotenuse is _____ times bigger than the shorter leg. The longer leg is √3 times bigger The hypotenuse is 2 times bigger than the shorter leg

  10. Just Watch Find the values of x and y. Give your answers in simplest radical form. 22 = 2x Hypotenuse = 2(shorter leg) 11 = x Divide both sides by 2. Substitute 11 for x.

  11. 60 20 x 30 y #1 You Try! 20=2x (hypotenuse is twice as big as shorter leg) 10=x Y=x√3 (long leg is √3 times bigger than shorter leg) Y=10√3

  12. Substitute for x. #2 You Try! Find the values of x and y. Give your answers in simplest radical form. Hypotenuse = 2(shorter leg) Divide both sides by 2. y = 27

  13. Just Watch Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. y = 2x Hypotenuse = 2(shorter leg). Simplify.

  14. x 60 30 #1 You Try! y 21=x√3 (long side is √3 larger than short side) X= 21/√3……21√3/3……..7√3 Y=2x (hypotenuse is twice as big as short side) Y=2(7√3) Y=14√3

  15. #2 You Try! Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. Hypotenuse = 2(shorter leg) x = 2y Simplify.

  16. Just Watch Find the values of x and y. Give your answers in simplest radical form. y = 2(5) y = 10 Simplify.

  17. 60 y 7 30 x You Try! X=7√3 (Long leg is √3 times larger than short side) Y=2(7) (Hypotenuse is twice as big as the short side) Y=14

  18. Special Triangles Summary Find the values of the variables. Give your answers in simplest radical form. 1.2. 3.4. x = 10; y = 20

  19. Finding an angle.(Figuring out which ratio to use and an inverse trig button.)

  20. 20 m ) 20 / 40 Tan-1 40 m Ex: 1 Figure out which ratio to use. Find x. Round to the nearest tenth. x Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem.

  21. 50 m 15 m ) 15 / 50 Sin-1 Ex: 2 Figure out which ratio to use. Find x. Round to the nearest tenth. x Shrink yourself down and stand where the angle is. Now, figure out which trig ratio you have and set up the problem.

  22. Ex. 3: Find . Round to the nearest degree. 17.2 9

  23. Ex. 4: Find . Round to the nearest degree. 7 23

  24. When we are trying to find a side we use sin, cos, or tan. When we are trying to find an angle we use (INVERSE) sin-1, cos-1,ortan-1.

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