Lessons 9.1 – 9.2
The Pythagorean Theorem & Its Converse
HW: Lesson 9.1 / 1-16 evens and
Lesson 9.2/1-16 evens
Essential Understanding
• Use the the Pythagorean Theorem to
solve problems.
• Use the Converse of the Pythagorean
Theorem to solve problems.
• Use side lengths to classify triangles by
their angle measures.
Pythagorean Theorem
If You Have A Right Triangle,
Then c²=a² + b²
c
a
b
The Pythagorean Theorem as some
students see it.
c
a
b
2
c =
2
a +
2
b
A better way
2
c
2
a
c
a
b
2
b
2
2
2
c =a +b
PYTHAGOREAN THEOREM
Applies to Right Triangles Only!
leg a
c
hypotenuse
b
leg
2
2
2
c =a +b
Pythagoras Questions
1
3 cm
x
4 cm
Pythagorean triple
2
x
5 cm
12 cm
Pythagorean triple
Pythagoras Questions: Finding a leg measure
3
xm
11m
9m
x ≈ 6.32 cm
Another method for finding a leg measure
4
23.8 cm
11
cm
x cm
x ≈ 21.11 cm
Applications of Pythagoras
1
6 cm
Find the diagonal of the rectangle
d
9.3 cm
d = 11.07 cm
A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm.
Find its perimeter.
2
4.3 cm
7.8 cm
x cm
x ≈ 6.51 cm
Perimeter = 2(6.51+4.3) ≈ 21.62 cm
therefore
The Converse Of The
Pythagorean Theorem
If c² =a² + b²,
Then You Have A Right Triangle
c
a
b
Do These Lengths Form Right Triangles?
i.e. do they work in the Pythagorean Theorem?
5, 6, 10
6, 8, 10
10² __5² + 6² 10²___6² + 8²
100___25 + 36 100___36 + 64
100≠ 61
NO
100 = 100
YES
Example of the Converse
Determine whether a
triangle with lengths 7,
11, and 12 form a right
triangle.
?
12  7  11
2
2
2
?
**The hypotenuse is
the longest length.
144  49  121
144  170
This is not a right triangle.
A Pythagorean Triple Is Any 3
Integers That Form A Right Triangle
3, 4, 5
5, 12, 13
Multiples Family
6,8,10
30,40,50
15,20,25
Multiples Family
10,24,26
25,60,65
35,84,91
Multiples of Pythagorean Triples are also
Pythagorean Triples.
Example of the Converse
Determine whether a
triangle with
lengths 12, 20, and
16 form a right
triangle.
?
20  12  16
2
2
2
?
400  144  256
400  400
This is a right triangle. A set of integers such
as 12, 16, and 20 is a Pythagorean triple.
Converse Examples
Determine whether
4, 5, 6 is a
Pythagorean triple.
4, 5, and 6 is not a
Pythagorean triple.
?
6 2  4 2  52
?
36  16  25
36  41
?
Determine whether
15, 8, and 17 is a
Pythagorean triple.
17 2  152  82
?
289  225  64
289  289
15, 8, and 17 is a Pythagorean triple.
Verifying Right Triangles
8
?
?
Note: squaring a square root!!
7
The triangle is a
right triangle.
Verifying Right Triangles
?
?
?
Note: squaring an integer & square root!!
15
36
The triangle is
NOT a right
triangle.
What Kind of Triangle??
You can use the Converse of the
Pythagorean Theorem to verify that a given
triangle is a right triangle or obtuse or acute.
What Kind Of Triangle ?
c² ?? a² + b²
Triangle Inequality
What Kind Of Triangle ?
c² ?? a² + b²
If the c²
= a² + b² , then right
If the c²
> a² + b² then obtuse
If the c²
< a² + b², then acute
The converse of the Pythagorean Theorem can be used to
categorize triangles.
Triangle Inequality
38, 77, 86
c2 ? a2 + b 2
862 ? 382 + 772
7396 ? 1444 + 5959
7396 > 7373
The triangle is obtuse
Triangle Inequality
10.5, 36.5, 37.5
c2 ? a2 + b 2
37.52 ? 10.52 + 36.52
1406.25 ? 110.25 + 1332.25
1406.24 < 1442.5
The triangle is acute
4,7,9
9²__4² + 7²
81__16 + 49
81 > 65 greater
OBTUSE
5,5,7
7² __5² + 5²
49 __ 25 +25
49 < 50 Less than
ACUTE
A Pythagorean Triple
3, 4, 5
25
9
5
3
2
5 =3 + 4
2
2
4
In a right-angled
triangle, the
square on the
hypotenuse is
equal to the sum
of the squares on
the other two
sides.
16
25=9 + 16
A 2nd Pythagorean Triple
5, 12, 13
169
25
5
13
12
In a right-angled
triangle, the
square on the
hypotenuse is
equal to the sum
of the squares on
the other two
sides.
144
2
2
2
13 =5 + 12
169=25 + 144
A 3rd
Pythagorean
Triple
49
2
2
625
7
25
24
2
25 =7 + 24
625=49 + 576
576
7, 24, 25
Building a foundation
• Construction: You use
four stakes and string to
mark the foundation of a
house. You want to
make sure the foundation
is rectangular.
a. A friend measures the
four sides to be 30 feet,
30 feet, 72 feet, and 72
feet. He says these
measurements prove that
the foundation is
rectangular. Is he
correct?
Building a foundation
• Solution: Your friend is not correct. The
foundation could be a nonrectangular
parallelogram, as shown below.
Building a foundation
b. You measure one of the diagonals to be
78 feet. Explain how you can use this
measurement to tell whether the
foundation will be rectangular.
Building a foundation
Solution: The diagonal
divides the foundation into
two triangles. Compare
the square of the length of
the longest side with the
sum of the squares of the
shorter sides of one of
these triangles.
• Because 302 + 722 =
782, you can conclude
that both the triangles are
right triangles. The
foundation is a
parallelogram with two
right angles, which
implies that it is
rectangular