Pythagorean Triples

1
Pythagorean Triples the Converse of the Theorem
Lesson 3.3.4
2
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
California Standard Measurement and Geometry
3.3 Know and understand the Pythagorean theorem
and its converse and use it to find the length of
the missing side of a right triangle and the
lengths of other line segments and, in some
situations, empirically verify the Pythagorean
theorem by direct measurement.
What it means for you Youll learn about the
groups of whole numbers that make the Pythagorean
theorem true, and how to use the converse of the
theorem to find out if a triangle is a right
triangle.

• Key words
• Pythagorean theorem
• Pythagorean triple
• converse
• right angle
• acute
• obtuse

3
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Up to now youve been using the Pythagorean
theorem on triangles that youve been told are
right triangles.
But if you dont know for sure whether a triangle
is a right triangle, you can use the theorem to
decide.
Its kind of like using the theorem backwards
and its called using the converse of the theorem.
4
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Pythagorean Triples are All Whole Numbers
You can draw a right triangle with any length
legs you like, so the list of side lengths that
can make the equation c2 a2 b2 true never
ends.
Most sets of side lengths that fit the equation
include at least one decimal thats because
finding the length of the hypotenuse using the
equation involves taking a square root.
5
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
There are some sets of side lengths that are all
integers these are called Pythagorean triples.
Youve seen a lot of these already.
For example
(3, 4, 5) (6, 8, 10) (8, 15, 17) (5, 12, 13)
6
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
You can find more Pythagorean triples by
multiplying each of the numbers in a triple by
the same number.
For example
To test if three integers are a Pythagorean
triple, put them into the equation c2 a2 b2,
where c is the biggest of the numbers. If they
make the equation true, theyre a Pythagorean
triple. If they dont, theyre not.
7
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Example 1
Are the numbers (72, 96, 120) a Pythagorean
triple?
Solution
To see if the numbers are a Pythagorean triple,
put them into the equation.
c2 a2 b2
Write out the equation
Substitute the values given
1202 722 962
14,400 5184 9216
Simplify the equation
14,400 14,400
which is true
These numbers are a Pythagorean triple.
Solution follows
8
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Guided Practice
Pythagorean triple
No, theyre not integers and they dont fit the
equation.
No, theyre not all integers.
Pythagorean triple
No, they dont fit the equation.
Pythagorean triple
Solution follows
9
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
The Converse of the Pythagorean Theorem
The Pythagorean theorem says that the side
lengths of any right triangle will satisfy the
equation c2 a2 b2, where c is the hypotenuse
and a and b are the leg lengths.
You can also say the opposite if a triangles
side lengths satisfy the equation, it is a right
triangle. This is called the converse of the
theorem
10
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Example 2
A triangle has side lengths 2.5 cm, 6 cm, and 6.5
cm. Is it a right triangle?
Solution
Put the side lengths into the equation c2 a2
b2, and evaluate both sides.
c2 a2 b2
Write out the equation
Substitute the values given
6.52 2.52 62
42.25 6.25 36
Simplify the equation
The longest side of a right triangle is the
hypotenuse, c.
42.25 42.25
which is true
It is a right triangle.
Solution follows
11
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Test Whether a Triangle is Acute or Obtuse
If a triangle isnt a right triangle, it must
either be an acute triangle or an obtuse
triangle.
Acute all angles lt 90
Obtuse one angle between 90 and 180
By seeing if c2 is greater than or less than a2
b2, you can tell what type of triangle it is.
12
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Example 3
A triangle has side lengths of 2 ft, 2.5 ft, and
3 ft. Is it right, acute, or obtuse?
Solution
Check whether c2 a2 b2, with c 3, a 2 and
b 2.5.
c2 32 9
Find the value of c2
a2 b2 22 2.52 4 6.25 10.25
Now find a2 b2
9 lt 10.25
Compare the two values
c2 lt a2 b2, so this is an acute triangle.
Solution follows
13
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Guided Practice
Right triangle
Acute triangle
Obtuse triangle
Right triangle
Obtuse triangle
Acute triangle
Solution follows
14
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Independent Practice
1. Every set of numbers that satisfies the
equation c2 a2 b2 is a Pythagorean triple.
Explain if this statement is true or not.
Not true a Pythagorean triple must be made up of
positive integers.
Solution follows
15
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Independent Practice
Pythagorean triple
Not
Not
Pythagorean triple
Not
Pythagorean triple
Solution follows
16
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Independent Practice
8. In triangle ABC, side AB is longest. If AB2
gt AC2 BC2 then what kind of triangle is ABC?
Triangle ABC is obtuse.
Solution follows
17
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Independent Practice
Are the following side lengths those of right,
acute, or obtuse triangles? 9. 5, 10, 14 10.
10, 11, 12 11. 12, 16, 20 12. 5.1, 5.3, 9.3 13.
2.4, 4.5, 5.1 14. 3.7, 4.7, 5.7
Obtuse triangle
Acute triangle
Right triangle
Obtuse triangle
Right triangle
Acute triangle
Solution follows
18
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Independent Practice
15. Justin is going to fit a new door. He
measures the width of the door frame as 105 cm,
the height as 200 cm, and the diagonal of the
frame as 232 cm. Is the door frame perfectly
rectangular?
No its sides and diagonal form an obtuse
triangle, not a right triangle.
Solution follows
19
Lesson 3.3.4
Pythagorean Triples the Converse of the Theorem
Round Up
The converse of the Pythagorean theorem is a kind
of backward version. You can use it to prove
whether a triangle is a right triangle or not
and if its not, you can say if its acute or
obtuse.