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Special Right Triangles-Section 9.7 Pages 405-412PowerPoint Presentation

Special Right Triangles-Section 9.7 Pages 405-412

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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 , and 2x, respectively.
- To prove this theorem we will need to setup a proof.

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

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.

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

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

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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