Unit 3 Study Guide Pythagorean Theorem 1.) Solving for side ‘c’ (hypotenuse – the longest side) You are missing the diagonal of the rectangle which creates a right triangle. a2 + b2 = c2 52 + 122 = c2 25 + 144 = c2 169 = c2 √169 = c 13 = c 2.) Solving for side ‘a’ or ‘b’ (legs) You are missing one of the legs of a right triangle. a2 + b2 = c2 a2 + 182 = 302 a2 + 324 = 900 -324 -324 a2 = 576 a = √576 a = 24 30 ft 18 ft 3.) Area of a square and Pythagorean Theorem The diagonal of a line is 20 feet. What is the approximate length of each side of the square? 20 Since the diagonal side is ‘c’, than c2 = 400. You need to find ‘a’ and ‘b’. What two same numbers squared would be closest to 400? 142 + 142 ≈ 400 196 + 196 ≈ 400 392 ≈ 400 4.) Solving for a missing distance on a coordinate plane Draw the length and width between the two points on the graph. These represent the two legs of the right triangle. Solve for side length ‘c’ by using a2 + b2 = c2 22 + 52 = c2 4 + 25 = c2 29 = c2 √29 ≈ c 5.4 ≈ c 5.) Pythagorean Triples a2 + b2 = c2 32 42 52 + = 9 + 16 = 25 A right triangle whose sides are all whole numbers. Examples. 3, 4, 5 5, 12, 13 6, 8, 10 7, 24, 25 6.) Pythagorean Theorem in 3d Figures Find the length of EO if EY = 20, YB = 24, and BO = 10. You need to find the hypotenuse (YO) of the bottom right triangle in order to use it as a leg in the 3D right triangle. a 2 + b2 = c 2 a2 + b2 = c2 102 + 242 = c2 202 + 262 = c2 100 + 576 = c2 400 + 676 = c2 Bottom 3D triangle 676 = c2 1076 = c2 Triangle √676 = c √1076 = c Length of EO 26 = c 33 ≈ c Volume of 3D Figures Cylinders, Cones, and Spheres Use 3.14 for п to solve for the volume of the following shapes. 1.) Cylinder (stack of circles) V = ∏r2h V = ∏ (32)(12) V = ∏ (9) (12) V = ∏ (108) V = 3.14 (108) V = 339.12 Plug in radius and height Square the radius first Multiply radius and height together Multiply by pi (3.14) 2.) Cone (1/3 the size of a cylinder) V = ∏r2h 3 V = ∏(102) (30) 3 V = ∏ (100) (30) 3 V = ∏ (3000) 3 V = ∏ (1000) Plug in radius and height Square the radius first Multiply radius and height together Divide by 3 V = 3,140 3.) Sphere V = 4∏r3 3 V = 4∏ (9)3 3 V = 4∏ (729) 3 V = 2916 ∏ 3 V = 972∏ V = 3,052.08 Plug in radius Cube the radius Multiply radius by 4 2916 is divisible by 3 so divide by 3 Multiply by pi (3.14) 4.) Volume of multiple shapes ~ Find total volume Cone Hemisphere (half sphere) 3 V = ∏(32) (6) 3 V = ∏(9)(6) 3 V = 54∏ 3 V = 18∏ V = 4∏r3 ÷ 2 3 V = 4∏(3)3 ÷ 2 3 V = 4∏(27) ÷ 2 3 V = 108∏ ÷ 2 3 V = 36∏ ÷ 2 V = 56.52 in3 V = 113.04 in3 ÷ 2 = 56.52 V= ∏r2h Cone + Hemisphere = 56.52 + 56.52 = 113.04 in3