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Understanding the Pythagorean Theorem and Special Right Triangles

This comprehensive guide explains the Pythagorean Theorem, illustrating how to find the lengths of the sides of right triangles. Learn about the relationship between the legs and the hypotenuse, and practice solving problems with step-by-step solutions. The guide also covers special right triangles, including the properties of 30-60-90 and 45-45-90 triangles. Master the relevant theorems and apply them in exercises to reinforce your understanding.

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Understanding the Pythagorean Theorem and Special Right Triangles

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  1. Pythagorean Theorem By: SAMUEL M. GIER

  2. DRILL • SIMPLIFY THE FOLLOWING EXPRESSION. 1. 4. + 2. 5. 3.

  3. REVIEW ABOUT RIGHT TRIANGLES A LEGS & The perpendicular side HYPOTENUSE B C The side opposite the right angle

  4. PYTHAGOREAN THEOREM • For any right triangle, the square of the length of the HYPOTENUSE is equal to the SUM of the squares of the length of the LEGS.

  5. ILLUSTRATION A HYPOTENUSE (AC)² = (AB)² +(BC)² LEG B C LEG

  6. ILLUSTRATIVE EXAMPLE Find AB, if AC = 15 and BC = 12. Solution 1: (AC)² = (AB)² +(BC)² (15)² = (AB)² +(12)² (225) = (AB)² + 144 225 - 144 = (AB)² 81 = (AB)² AB = AB = 9 A HYPOTENUSE LEG B C LEG (Squaring both sides)

  7. ILLUSTRATIVE EXAMPLE Find BC, if AC = 20 and AB = 12. Solution 1: (AC)² = (AB)² +(BC)² (20)² = (12)² +(BC)² 400 = 144 +(BC)² 400 - 144 = (BC)² 256 = (BC)² BC = BC = 16 A HYPOTENUSE LEG B C LEG (Squaring both sides)

  8. ILLUSTRATIVE EXAMPLE Find BC, if AC = 20 and AB = 12. Solution 2: (BC)² = (AC)² - (AB)² (BC)² = (20)² - (12)² (BC)² = 400 - 144 (BC)² = 256 BC = BC = 16 A HYPOTENUSE LEG B C LEG (Squaring both sides)

  9. EXERCISES 1. • Find the value of k. Solution 1: Write an equation for k (k)² = (15)² +(20)² k² = 225 +400 k² = 625 k = k = 25 15 20 k (Squaring both sides) Answer : 25

  10. EXERCISES 2. • Find the value of k. Solution 1: Write an equation for k (14)² = (k)² +(6)² 196 = k² +36 196-36 = k² 160= k² =k k = 4 k 6 14 Answer: 4

  11. EXERCISES 2. • Find the value of k. Solution 2: Write an equation for k (k)² = (14)² -(6)² k² = 196 - 36 k² = 160 k = k = 4 k 6 14 Answer: 4

  12. EXERCISES 3. • Find the value of k. Solution 1: Write an equation for k (k)² = (13)² -(5)² k² = 169 - 25 k² = 144 k = k= 12 5 h 13 Answer: 12

  13. SPECIAL RIGHT TRIANGLES • 30 – 60 – 90 TRIANGLE • 45 – 45 – 90 TRIANGLE (Isosceles Right Triangle) 30º 45º 60º 45º

  14. SPECIAL RIGHT TRIANGLES • THEOREM “In a 45 -45 -90 triangle, the length of the hypotenuse is equal to the SQUARE ROOT of the LEG. • 45 – 45 – 90 TRIANGLE (Isosceles Right Triangle) 45º 45º

  15. SPECIAL RIGHT TRIANGLES • Hence, the legs of an Isosceles Right Triangle are equal. So, either equation is true. • THEOREM “In a 45 -45 -90 triangle, the length of the hypotenuse is equal to the SQUARE ROOT of the LEG. c = a c a a c = b 45º c b 45º b

  16. Examples • 1. Find the value of c. • 2. Find the value of z 45º z 7 c Solution: c = a c = 5 45º 5 SOLUTION: z = a =(7 )( ) = 7 = 7 ( 2) Z = 14

  17. SPECIAL RIGHT TRIANGLES • 30 – 60 – 90 TRIANGLE • THEOREM “ In a 30- 60- 90 triangle, the side opposite the 30º is equal to ONE – HALFthe length of the HYPOTENUSE.” THUS, Shorter leg = ½ (hypotenuse) Or a = ½ c 30º LONGER LEG HYPOTENUSE (c) ( b) 60º SHORTER LEG (a)

  18. SPECIAL RIGHT TRIANGLES • 30 – 60 – 90 TRIANGLE • THEOREM “ In a 30- 60- 90 triangle, the side opposite the 60º is equal to the length of the SHORTER LEG times the square of 3.” THUS, longer leg = (shorter leg) x Or b = a 30º HYPOTENUSE LONGER LEG (c) ( b) 60º SHORTER LEG (a)

  19. Summary 1. Shorter leg = ½ (hypotenuse) Or a = ½ c 2. Hypotenuse =( 2 ) x ( shorter leg) Or C = 2a 3. longer leg = (shorter leg) x Or b = a • 30 – 60 – 90 TRIANGLE 30º LONGER LEG HYPOTENUSE (c) ( b) 60º SHORTER LEG (a)

  20. In the figure, find the value of c and b. SOLUTION: 1. TO FIND c. ( side Opposite 90º) c= 2a = 2( 5) C = 10 2. TO FIND b. (side Opposite 60º) b = a = 5 30º c b 60º a =5

  21. In the figure, find the value of x and y. SOLUTION: 1. TO FIND y. ( side Opposite 30º) y = ½ ( 20) y = 10 2. TO FIND x. (side Opposite 60º) x = y = 10 30º 20 x 60º y

  22. QUIZ • 2. Find the value of a and b. • 1. Find the value of x. 30º 45º x 16 b 7 60º a

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