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>.