111'24' Mathematics, Grade 8'

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Texas Essential Knowledge and Skills (TEKS)
111.24. Mathematics, Grade 8.
(8.7)  Geometry and spatial reasoning. The
student uses geometry to model and
describe the physical world. The student is
expected to (C)  use pictures or models to
demonstrate the Pythagorean Theorem (8.9)  Measur
ement. The student uses indirect measurement to
solve problems. The student is expected
to (A)  use the Pythagorean Theorem to solve
real-life problems and (B)  use proportional
relationships in similar two-dimensional figures
or similar three-dimensional figures to find
missing measurements.
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Solving Right Triangles
Steps to find the angles and length of the sides
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Solving Right Triangles

• Rules for Special Triangles
• Pythagorean Theorem
• Trigonometric ratios

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Before We Start
Lets Review the Basics
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What is a Triangle ?

• One of the basic shapes of geometry a polygon
  with three vertices and three sides which are
  straight line segments.
• Triangles ALWAYS have 3 sides. (tri- means 3).
• 3) The interior angles of a ? add up to 180
  degrees.
• 4) Four (4) types of triangles
• Equilateral Triangle
• Isosceles Triangle
• Right- Angled Triangle
• Scalene Triangle

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Equilateral Triangle
3 sides of equal length 3 angles that are
equal. Since ALL the angles in a triangle add
up to 180º then 180 divided by 3 must be 60º.
60
60
60
The clue is in the name EQUILateral.
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Isosceles Triangle
2 sides of equal length 2 angles that are
equal.
Sometimes called an Acute ?. All angles are
less than 90O
X
70
70
Using the above example All angles in a triangle
add up to 180 so 180 - (7070) ? x
40
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Scalene Triangle
NO sides of equal length NO angles that are
equal. But ALL the angles in this ? still add
up to 180º.
Sometimes called an Obtuse ?. One angle is
greater than 90O
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Right Triangle
Contains a right angle (an angle of 90)
The other two angles add up to 90
The side opposite the right angle is the
Hypotenuse
90
Because all the angles in a triangle add up to
180 a Right Angled Triangle must have one
angle of 90.
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Rules for Special Triangles

• 45-45-90 Triangle
• 30-60-90 Triangle

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45-45-90 Triangle
B
xv2
x
A
C
x
the measure of the hypotenuse is equal to the
measure of a leg (x) multiplied by v2
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30-60-90 Triangle
B
60
c 2x
a x
30
C
A
b xv3

• the measure of the hypotenuse (leg c) is two
  times the length of the leg opposite the 30O
  angle (leg a) (answer 2x).
• The measure of the other leg (leg b) is v3 times
  that of the leg opposite the 30O angle (leg a).
  (answer xv3 )

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The Pythagorean Theorem

• Where did it come from ?
• What is that?

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Who was Pythagoras

• Pythagoras of Samos
• Born approximately 582 BC507 BC
• A Greek mathematician and philosopher.
• Best known for the Pythagorean theorem which
  bears his name.
• Known as The Father of Numbers
• Pythagoras and his students believed that
  everything was related to mathematics.

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The Pythagorean Theorem
is
a2 b2 c2
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When working with any right ?
The square of (leg a)
plus the square of (leg b)
is equal to the square of the hypotenuse (leg c)
a2
b2
c2
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When Solving Right Triangles
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B
Hypotenuse
c
a
A
C
b
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Example1
B
c
2v10
2
A
C
6
a2 b2 c2
22 62 c2
4 36 c2
c2 40
c v40? v4x10 ? 2v10
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Example2
B
c
5v5
5
A
C
10
a2 b2 c2
52 102 c2
25 100 c2
125 c2
c v125 ? v5x25 ? 5v5
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Example3
B
8
4
A
C
b
4v3
a2 b2 c2
42 b2 82
16 b2 64
b2 48
b v48 ? v16x3 ? 4v3
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Review What We Have Learned
Triangles ALWAYS have 3 sides
The interior angles of a ? add up to 180 degrees.
Four (4) types of triangles Equilateral
Triangle Isosceles Triangle Right-
Angled Triangle Scalene Triangle
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Review - Continued

• 2 Types of Right ?s have special rules
• 30 60 90
• 45 45 90

All right ?s can use Pythagorean Theorem to solve
a2 b2 c2
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Homework
Worksheet XX XX
Due Tomorrow