Lesson 3: Introduction to Graphing
Digging Deeper solutions
The Pythagorean Theorem states that if a triangle is a right triangle with legs
a and b, and hypotenuse c, then a2 + b2 = c 2 . Any set of integers a, b, and c
which satisfy the Pythagorean Theorem are called “Pythagorean triples” and
are usually written as (a, b, c). Note: We usually write the legs beginning
with the smaller value.
In the table below, there are several Pythagorean triples. The lengths of the
legs have been given; you are expected to find the hypotenuse. (The first
one has been done for you.) As you find the values, look to see what
patterns you notice.
Note: There are extra rows at the bottom in case you want to keep
generating Pythagorean triples.
a
b
c
a2 + b2 = c 2
3
4
5
32 + 42 = c 2 ; 9 + 16 = c 2 ; 25 = c 2 ; 5 = c
6
8
10
62 + 82 = c 2 ; 36 + 64 = c 2 ; c 2 = 100 ; c = 10
9
12
15
92 + 122 = c 2 ; 81 + 144 = c 2 ; c 2 = 225 ; c = 15
12
16
20
122 + 162 = c 2 ; 144 + 256 = c 2 ; c 2 = 400 ; c = 20
15
20
25
152 + 202 = c 2 ; 225 + 400 = c 2 ; c 2 = 625 ; c = 25
Write a letter to another student explaining the patterns you noticed above.
If you start with a Pythagorean triple, you can multiply each of the
numbers by the same number, and the new group will also be a
Pythagorean triple. For example, if you multiply the Pythagorean triple
(3, 4, 5) by 10, you will get (30, 40, 50), which is also a Pythagorean
triple.
Algebra 1
© 2009 Duke University Talent Identification Program
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Lesson 3: Introduction to Graphing
Digging Deeper solutions
Use the patterns you noticed from above to find the value of c when a=63
and b= 84. Note: Do not use the Pythagorean theorem to solve this
problem. You should be able to find the answer by using the pattern from
above.
Since a = 63 = 3 ⋅ 21 and b = 84 = 4 ⋅ 21 , then the hypotenuse is
c = 5 ⋅ 21 = 105 .
Check:
?
a2 + b2 = c 2
?
632 + 842 = 1052
?
3969 + 7056 = 11025
?
11025 = 11025 On the previous page, we started with the Pythagorean triple (3, 4, 5). This
is called a “primitive Pythagorean triple” because its values are relatively
prime. (Relatively prime means that the GCF of all the numbers is 1. In this
case, the GCF of 3, 4 and 5 is 1 since 1 is the only number that divides each
of the numbers.)
The triples (5, 12, 13), (8, 15, 17), and (7, 24, 25) are also primitive
Pythagorean triples. (If you want, you can use the Pythagorean theorem to
verify that in each case a2 + b2 = c 2 .)
Use the pattern you discovered from above to fill in the table below for each
of the Pythagorean triple families. Do Not use the Pythagorean theorem—
the purpose of this activity is to quickly generate Pythagorean triples. Note:
The numbers in the last row are not true Pythagorean triples since they are
not all integers, but the pattern still applies.
a
b
c
a
b
c
a
b
c
5
12
13
8
15
17
7
24
25
10
24
26
24
45
51
168
576
600
35
84
91
160
300
340
49
168
175
55
132
143
64
120
136
35
120
125
Algebra 1
© 2009 Duke University Talent Identification Program
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Lesson 3: Introduction to Graphing
Digging Deeper solutions
5
7
12
7
13
7
16
15
2
34
15
1
24
7
25
7
Solve the following problems. Find the solution using Pythagorean triples
instead of the Pythagorean theorem.
1. Find x
36
15
x
The legs (15 and 36) have a GCF of 3. When we divide both legs
by 3, we get 5 and 12. This triangle is a multiple of the (5, 12,
13) Pythagorean triple. The multiplier is 3, so the hypotenuse is
3 ⋅ 13 = 39 .
Check:
?
152 + 362 = 392
?
225 + 1296 = 1521
?
1521 = 1521 Answer: x = 39
2. How far up a wall will a 25-ft ladder reach, if the base of the ladder is 7
feet from the wall?
25
d
7
Algebra 1
© 2009 Duke University Talent Identification Program
Page 3 of 4
Lesson 3: Introduction to Graphing
Digging Deeper solutions
The lengths 7 and 25 are relatively prime. So this is a primitive
Pythagorean triple (7, 24, 25). The ladder will reach 24 feet up
the wall.
3. A right triangle has legs of length 2 and 3¾. What is the length of the
hypotenuse? (Try to do this using the patterns associated with
Pythagorean triples.)
15
8
and 2 = . This is the Pythagorean triple (8, 15, 17) with
4
4
1
17
a multiplier of . The length of the hypotenuse is
.
4
4
3 34 =
Check:
2
2
 15  ?  17 
22 + 
 =

 4   4 
225 ? 289
4+
=
16
16
?
64 225 289
+
=
16
16
16
?
289 289
=
16
16
Answer: Length of hypotenuse is
Algebra 1
© 2009 Duke University Talent Identification Program
Page 4 of 4
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