Special Right Triangles-Section 9.7Pages 405-412

Adam DecSection 830 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.

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The Proof Given: Triangle ABC is equilateral, ray BD bisects angle ABC.

Prove: DC: DB: CB= x: x : 2x3

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

6090

2x

x

2 2 2

2 2 2

2 2

3

3

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.

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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

2

x

x

2 2 2

2 2

2

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The Easy Problems

The Moderate Problems

The Difficult Problems

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

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3 3

3

2

3

3

2

3 6

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Works CitedRhoad, 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>.