Transcript Slide 1

Triangle
Properties
Distance
Congruent
Similar
Pythagorean
Trigonometry
Transit – measure
Use angles and distance
Similar Triangles
Congruent Triangles
Right Triangles
and the
Pythagorean Theorem
Right Triangles
and the
Pythagorean Theorem
Escher used right
angles to form his
artwork known as
Waterfall.
Contains
three uses of the impossible tribar
Created by
British mathematician
Roger Penrose (b 1931) in 954.
Pythagorean Theorem
2
a
+
2
b
=
2
c
Pythagoras of Samos
Greek mathematician
about 569 BC about 475 BC
Pythagorean Theorem
In any right angled triangle:
• The area of the square whose side is the
hypotenuse
• =
• The sum of the areas of the squares whose sides
are the two legs
The theorem can be written as an equation
relating the lengths of the sides a, b and c,
often called the Pythagorean equation
a2 + b2 = c2
Given a right triangle
Pythagorean Theorem
uses relationship of side lengths
1.c is the longest side of the triangle,
called the "hypotenuse"
2.a and b are the legs
Pythagorean Theorem
Pythagorean Theorem
2
a
+
2
b
=
2
c
President
Garfield’s
Proof
1881
One uses subtraction,
the other addition.
How can opposite
operations prove the
same thing?
2
a +
2
b
2
=c
Algebraic Proof
Area of Large Square
A = (a+b) (a+b)
Area of Pieces
A=c*c
A = 4 (1/2) ab
Total Area of Pieces
A = c2 + 2ab
Both areas are equal
Therefore,
(a+b) (a+b) = c2 + 2ab
a2+ + 2ab + b2 =
a2+b2 = c2
Similarity of the triangles leads to the equality of ratios
of corresponding sides:
The first result equates the cosine of each angle θ and the second result
equates the sines. These ratios can be written as:
BC2 = AB x BH and AC2 = AB x AH
BC2 + AC2 = ABxBH + ABxAH = AB (AH + BH) = AB2
The Pythagorean Theorem
BC2 + AC2 = AB2
Pythagorean Theorem
Given Points R (-4,5) and S (2, -1)
Determine the length of Line RS.
How high
up the wall
does the
ladder
reach?
Broken Pole
Napoleon’s Officer
Given ∠DEG and ∠DEF are right angles
∠EDG
∠EDF
Prove
EF EG
Solve for x.
Solve for x.
13’
12’
7’
x’
Solve for x’.
Pythagorean and a bit more
• A diver hooked to a 25meter line is searching for
the remains of a Spanish
galleon in the Caribbean
Sea. The sea is 20 meters
deep and the bottom is
flat.
• What is the area of the circular region that the
diver can explore?
Pythagorean and a bit more
• One of the sketches below shows the greatest
area that you can enclose in a right angled
corner with a rope of length s.
• Which one? Explain your reasoning.
Rationalize the Denominator
• When you move a root out from the
denominator of a fraction to the numerator
• Equations are often not considered
'completely simplified' if there is still a radical
in the denominator
• The denominator should not have an
irrational number
• Fix by making the denominator rational
• Called "Rationalizing the Denominator"
45-45 Triangle
45
a2 + b2 = c2
x2 + x 2 = c 2
2x2 = c2
IF x = 1
2 = c2
X
2 c
45
90
X
Hippasus of Metapontum
30-60 triangle
Pythagorean Triple
•Set of 3 positive whole numbers
•a, b, c – side lengths of a right triangle
•a, b, c – satisfy the equation from the
Pythagorean Theorem a2 + b2 = c2
Table of the first few primitive Pythagorean triples:
r
s
a
b
c
2
1
3
4
5
3
2
5
12
13
4
1
15
8
17
4
3
7
24
25
5
2
21
20
29
5
4
9
40
41
6
1
35
12
37
6
5
11
60
61
7
2
45
28
53
Pythagorean triple scatterplot
Plot of triples generated by Euclid's formula
Pythagorean Inequalities
The following triples represent the side lengths
of a triangle. Determine whether the
triangle is right, acute or obtuse.
Obtuse
1. 5, 9, 12
Obtuse
2. 7, 24, 26
Acute
3. 13, 15, 17
Right
4. 3, 4, 5
Acute
5. 25, 25, 30
1. You've just picked up a ground ball at first
base, and you see the other team's player
running towards third base. How far do you
have to throw the ball to get it from first base
to third base, and throw the runner out?
2. You're locked out of your house and the only open
window is on the second floor, 25 feet above the ground.
You need to borrow a ladder from one of your neighbors.
There's a bush along the edge of the house, so you'll have
to place the ladder 10 feet from the house.
Your neighbor has a ladder that is 14’ long. Will you
borrow the ladder?
a
c
b
c1
b1
a1
THE GENERAL EXTENSION TO PYTHAGORAS' THEOREM:
If any 3 similar shapes are drawn on the sides of a right
triangle, then the area of the shape on the hypotenuse
equals the sum of the areas on the other two sides.
But why?
Pythagorean Work in Music
• Music of the Spheres
• Harmony through the
ratio of string lengths
Solve for the
Unknown
Pythagorean Theorem
Spiral of Theodorus
Spiral of
Theodorus
to 3 windings
Geometry in Art
Spherical Geometry
Hyperbolic Geometry
Parallax
Parallax
parallax angle = half of total angle
L=
half of total baseline
-------------------------------------tan [ half of total angle]
Similar Triangles
Transit of Venus
It takes about 7 hours for Venus to
cross the Sun.
Photos taken during the transit
show Venus following its orbit
around the Sun.
To transit means to
pass across or
through somewhere.
The tiny shadow of
Venus on the Sun
transits across the
sun's disc.
Why are Transits of Venus so rare?
Simulation of the parallax of a star which is at a distance of 1 pc
(closer to our Sun than any real star) Subtle change January to July