Download File - Mr. Heyboer`s Geometry

CHAPTER 8: RIGHT TRIANGLES
• Watch the follow clip about the Sunshine Skyway
Bridge. Think about the design and construction. Do
you recognize any right triangles?
INTEGERS
• Integer: One of a set of positive and negative
whole numbers including zero.
• -3, -2, -1, 0, 1, 2, 3…
• When using measurements:
• Positive Integers
N:
1
2
3
4
5
6
7
8
9
N2
1
4
9
16
25
36
49
64
81
Take a look at three squares: 9, 16, and 25. Note that 9+16=25, or
32 + 42 = 52
MANIPULATING EQUATIONS
• 32 + 42 = 52 Is an equation.
• Can we find more numbers among the squared integers such
that the sum of the two smaller squares is equal to the largest
square?
• Because we have an equation, we can simply multiply each
side of the equation by the same number to get another
equation.
• 32 + 42 = 52
• (3*2)2 +(4*2)2 = (5*2)2
62 + 82 = 102
36 + 64 = 100 TRUE
PYTHAGOREAN TRIPLES
• Integers a, b, and c form a Pythagorean Triple if
a2 + b2 = c2, where a and b are the smaller numbers
and c is the largest.
• Take 5, 12, and 13. How can we tell if they are
Pythagorean Triples? PLUG IT IN! 
• Does 52 + 122 = 132 ?
• 25+144= 169 ?
• 169 = 169 YES, therefore they are Triples.
• Remember, more triples can be created by
multiplying each integer in the equation by the
same number.
PYTHAGOREAN TRIPLES
• Is (4, 5, 6) a Pythagorean Triple?
•
•
•
•
4 and 5 are the smaller integers; 6 is the largest
42 + 5 2 = 62
16+25=36 FALSE.
(4, 5, 6) is NOT a Pythagorean Triple.
PRACTICE
• Take out personal whiteboards and determine
whether each set of integers provided is a
Pythagorean Theorem.
• Keep notes out for later.
PLATO’S FORMULA
Plato provided us with many awesome
ideas. One of them is a way to
mathematically calculate many of the
Pythagorean Triples. Don’t you wish you
could have hung out with him?
For any positive integer, m:
(2m)2 + (m2 – 1)2 = (m2 +1)2
PLATO’S FORMULA
(2m)2 + (m2 – 1)2 = (m2 +1)2
• Use Plato’s Formula for m=2. Check to see if your
answer is a Pythagorean Triple.
• (2*2)2 + (22 – 1)2 = (22 +1)2
• (4)2 + (3)2 = (5)2
• 16+9=25 TRUE.
• Plato’s Formula shows that (3,4,5) is a Pythagorean
Triple.
PRACTICE
(YOU KNOW YOU LOVE IT)
• Use Plato’s formula to find Pythagorean Triples for the
following integers. Use a calculator if necessary.
• m=6 (display on whiteboard when done)
• Answer: (12, 35, 37)
• m=3 (display on whiteboard when done)
• Answer: (6, 8, 10)