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Special Right Triangles-Section 9.7 Pages 405-412

Special Right Triangles-Section 9.7 Pages 405-412. Adam Dec Section 8 30 May 2008. Introduction. Two special types of right triangles. Certain formulas can be used to find the angle measures and lengths of the sides of the triangles.

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Special Right Triangles-Section 9.7 Pages 405-412

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  1. Special Right Triangles-Section 9.7Pages 405-412 Adam Dec Section 8 30 May 2008

  2. Introduction • Two special types of right triangles. • Certain formulas can be used to find the angle measures and lengths of the sides of the triangles. • One triangle is the 30-60-90(the numbers stand for the measure of each angle). • The second is the 45-45-90 triangle.

  3. 30- 60- 90 • 30 - 60 - 90 - Triangle Theorem: In a triangle whose angles have measures 30, 60, and 90, the lengths of the sides opposite these angles can be represented by x, x , and 2x, respectively. • To prove this theorem we will need to setup a proof.

  4. The Proof Given: Triangle ABC is equilateral, ray BD bisects angle ABC. Prove: DC: DB: CB= x: x : 2x Since triangle ABC is equilateral, Angle DCB= 60, Angle DBC= 30 , Angle CDB= 90 , and DC= ½ (BC) According to the Pythagorean Theorem, in triangle BDC: x + (BD) = 2x x + (BD) = 4x (BD) = 3x BD = x Therefore, DC: DB: CB= x: x : 2x 30 2x 90 60 x

  5. 45- 45- 90 • 45 - 45 - 90 - Triangle Theorem: In a triangle whose angles have measures 45, 45, 90, the lengths of the sides opposite these angles can be represented by x, x, x , respectively. • A proof will be used to prove this theorem, also.

  6. The Proof Given: Triangle ABC, with Angle A= 45 , Angle B= 45 . Prove: AC: CB: AB= x: x: x Both segment AC and segment BC are congruent, because If angles then sides( Both angle A and B are congruent, because they have the same measure). And according to the Pythagorean theorem in triangle ABC: x + x = (AB) 2x = (AB) X = AB Therefore, AC: CB: AB= x: x: x x x

  7. The Easy Problems

  8. The Moderate Problems

  9. The Difficult Problems

  10. The Answers • 1a: 7, 7 ; 1b: 20, 10 ; 1c: 10, 5; 1d: 346, 173 ; 1e: 114, 114 • 5: 11 • 17a: 3 ; 17b: 9; 17c: 6 ; 17d: 1:2 • 21a: 48; 21b: 6 + 6 • 25a: 2 + 2 ; 25b: 2 • 27: [40(12 – 5 )] 23

  11. Works Cited Rhoad, Richard. Geometry for Enjoyment and Challenge. New. Evanston, Illinois: Mc Dougal Littell, 1991. "Triangle Flashcards." Lexington . Lexington Education. 29 May 2008 <http://www.lexington.k12.il.us/teachers/menata/MATH/geometry/triangl esflash.htm>.

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