Special Right Triangles- Section 9.7, Pg 405412
Download
Report
Transcript Special Right Triangles- Section 9.7, Pg 405412
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>.