Agenda 2/27 1) Bell Work 2) IP Check 3) Outcomes 4) Pythagorean Proof #2 from 9.2 Notes 5) Special Right Triangles – A) – B)

Bell Work 2/27/12 1) Use the Geometric Mean Theorems to find the value of x. A)B) 2) Use Pythagorean Theorem to find the value of x. A)B)

Outcomes I will be able to: 1) Use Pythagorean Theorem 2) Recognize special right triangles, and ways to solve for the legs and hypotenuse in them

Proof #2 9.2 Notes from Friday (a + b)(a + b) c² + 4 x (1/2)(a x b) Looking at just the big square: Looking at little square and 4 triangles: (a + b)(a + b) = c² + 4 x (1/2)(a x b) Use FOIL: a² + 2ab + b² = c² + 2ab -2ab -2ab a² + b² = c²

Notes 9.4 Exercise: In the space below is an equilateral triangle. Label its vertices with A, B, and C, and draw in an altitude CD. 1. What are the measures of A and B? _______ What are ACD and BCD? _________ What are ADC and BDC? ___________ 2. Is ∆ADC congruent to ∆BDC? ______ Why? ________________________ 3. Is ? Why? How do AC and AD compare? In a 30º-60º-90º triangle: ____________________________________________________________. A B C D Yes SAS The hypotenuse is always twice the shorter leg

Notes Let AD = x and AC = 2x. Use the Pythagorean Theorem to find CD in terms of x. CD² + x² = (2x)² CD² + x² = 4x² - x² - x² CD² = 3x² CD = x x 2x

Special Right Triangle Theorems a b c 2 x See examples 3, 4, 5, 7, 8

Special Right Triangle Theorems ***See examples 1, 2, 6

Examples Examples: 1. In a 45°-45°-90° triangle, let the legs equal 6 and the hypotenuse equal x. Solve for x. 6 6 x

Examples 2. In a 45°-45°-90° triangle, let the legs equal x and the hypotenuse equal 10. Solve for x. 10 x x

Examples 3. In a 30°-60°-90° triangle, let the smallest leg be 20. Find the lengths of the other two sides. ***Use the Triangle Theorem: 20 l h

Examples 4. In a 30°-60°-90° triangle, the hypotenuse is 50. Find the lengths of the other two sides. 50 s l

Examples

7. A ramp is used to unload trucks. How high is the end of a 50 inch ramp when it is tipped by a 30° angle? in x

Examples 8. The roof of a doghouse is shaped like an equilateral triangle with height 3 feet. Estimate the area of the cross-section of the roof. 3 ft

Special Right Triangle Rules : Different Methods depending upon what we are given: If we are given the short leg: To find the long leg, multiply the short leg by √3 So, b = 3√3 To find the hypotenuse multiply the short side by 2 So, c = 2 x 3 = 6

Special Right Triangles If we are given the long leg: ***Always find the short leg first by dividing by √3, so a = To find the hypotenuse, multiply the short side by 2, so c =

Special Right Triangle Rules If we are given the hypotenuse: ***Find the short leg first, by dividing the hypotenuse by 2, a = 10÷2=5 Now, multiply the short leg by √3 to find the long leg, b = 5 x √3 = 5√3

Special Right Triangles Rules : The hypotenuse of a right triangle will always be a multiple of the √2. The legs will always be congruent. ***To find the legs, simply divide the hypotenuse by √2