Transcript Section 9.6 Families of Right Triangles

Section 9.6
Families of Right Triangles
By: Maggie Fruehan
Any three whole numbers that satisfy the equation
a²+b²=c² form a Pythagorean Triple.
15
6
5
3
8
15
12
10
4
20
25
9
These four triangles are all members of the (3,4,5) family.
Some History…
The study of these Pythagorean triples began long before the
time of Pythagoras.
There are Babylonian tablets that contain lists of such triples.
Pythagorean triples were also used in ancient Egypt. For
example, to produce a right angle they took a piece of string,
marked it into 12 equal segments, tied it into a loop, and held
it taut in the form of a (3,4,5) triangle.
The number
of spaces
match the
(3,4,5) triple!
String with 12 knots
Sting pulled taut
Pythagorean Triples must appear as whole numbers.
3 3
2½
1½
2
4 3
0.5
5 3
0.4
0.3
Even though these are not families, they all are
members of the (3,4,5) family.
There are infinitely many families, but the most
frequently seen are the:
(3,4,5)
45
?
27
72
(5,12,13)
45/9= 5
27/9= 3
? /9= 4
72/6= 12
30/6= 5
30
?
? =36
? =78
? /6= 13
2.5(10)= 25
(7,24,25)
2.5
0.7
?
0.7(10)= 7
? (10)= 24
? = 2.4
More Families…
1½
1½ =
(8,15,17)
?
2
2=
?=
3
2
4
2
5
2
18/2= 9
(9,40,41)
?
18
80
Numerators
resemble the
(3,4,5)
triangle!
80/2= 40
?/ 2= 41
?=
1
2
2
? = 82
The Principal of the Reduced Triangle
 Reduce the difficulty of the problem by multiplying or
dividing the three lengths by the same number to obtain
a similar, but simpler, triangle in the same family.
 Solve for the missing side of the easier triangle.
 Convert back to the original problem.
x
4
2x
2(4)=8
7½
2(7½)=15
The family
is (8,15,17).
Thus,
2x=17 and
x=8½ (in
the original
problem).
More Reduced Triangles!
 200
4 + y² = 9
600
400
2² + y² = 3²
*Make sure to
change the
variable!
3
2
y
x =200 5
You can also enlarge the triangle!
1¼
y² = 5² + 8²
y
y² = 89
y = 89
5
2
4
y= 5
 200
4
x
y² = 5
8
15
Find x.
17
25
6
26
x=7
x
x =12
15
13
Find x
13
x
25
Find x
41
x
40
41
Square
x = 18 2
HA!!! IT’S NOT 10!!!
because it’s a…
6² + x² = 8²
36 + x² = 64
x² = 28
8
6
x = 28
x=2 7
x
http://www.itsatrap.net/
Works Cited
 Richard
Rhoad, George Milauskas,
Robert Whipple, Geometry for
Enjoyment and Challenge. Evantson,
Illinois: McDougal, Littell & Company,
1991.
 http://www.math.brown.edu/~jhs/frintc
h2ch3.pdf
 http://itsatrap.net/