Special Right Triangles-Section 9.7 Pages 405-412

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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 9.7Pages 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