* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Special Right Triangles- Section 9.7, Pg 405412
Survey
Document related concepts
Golden ratio wikipedia , lookup
Noether's theorem wikipedia , lookup
Euler angles wikipedia , lookup
History of geometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Incircle and excircles of a triangle wikipedia , lookup
Euclidean geometry wikipedia , lookup
Transcript
Special Right TrianglesSection 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. One triangle is the 30-60-90(the numbers stand for the measure of each angle). The second is the 45-45-90 triangle. 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 3, and 2x, respectively. To prove this theorem we will need to setup a proof. Given: Triangle ABC is equilateral, ray BD bisects angle ABC. The Proof Prove: DC: DB: CB= x: x 3: 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: 30 2 2 2 2 2 2x x + (BD) = 2x x 2 + (BD) 2 = 4x 2 (BD) = 3x 90 60 BD =x 3 x Therefore, DC: DB: CB= x: x 3 : 2x 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 2 , respectively. A proof will be used to prove this theorem, also. The Proof Given: Triangle ABC, with Angle A= 45 , Angle B= 45 . Prove: AC: CB: AB= x: x: x x 2 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: 2 x + x 2 = (AB) x 2 2x 2 = (AB) 2 X 2 = AB Therefore, AC: CB: AB= x: x: x 2 The Easy Problems The Moderate Problems The Difficult Problems The Answers 1a: 7, 7 3 ; 1b: 20, 10 3; 1c: 10, 5; 1d: 346, 173 3 ; 1e: 114, 114 3 5: 11 2 17a: 3 3; 17b: 9; 17c: 6 3; 17d: 1:2 21a: 48; 21b: 6 + 6 2 25a: 2 + 2 3 ; 25b: 2 6 27: [40(12 – 5 3 )] 23 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>.