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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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## Special Right Triangles-Section 9.7Pages 405-412

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 , 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 Difficult Problems

• 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/trianglesflash.htm>.