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.
If You Have A Right Triangle,If You Have A Right Triangle,Then Then c²=c²=aa² + b² ² + b²
Pythagorean Theorem
aa
bb
cc
The Pythagorean Theorem as some students see it.
ac
b
c2 = a2 + b2
A better way
a2
b2
c2
c
b
a
c2=a2+b2
Applies to Right Triangles Only!
leg
leg
hypotenusea
b
c
PYTHAGOREAN THEOREMPYTHAGOREAN THEOREM
c2=a2+b2
3 cm
4 cm
x
1
5 cm
12 cm
x2
Pythagoras Questions
Pythagorean triple
Pythagorean triple
x m
9 m
11m
3
11 cm
x cm
23.8 cm
4
Pythagoras Questions: Finding a leg measure
x ≈ 6.32 cm
x ≈ 21.11 cm
Another method for finding a leg measure
Applications of Pythagoras
Find the diagonal of the rectangle
6 cm
9.3 cm
1
d
d = 11.07 cm
A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm. Find its perimeter.
2
7.8 cm
4.3 cm
x cm
Perimeter = 2(6.51+4.3) ≈ 21.62 cm
x ≈ 6.51 cm
therefore
The Converse Of The The Converse Of The Pythagorean TheoremPythagorean Theorem
If If c² =a² + b²c² =a² + b², , ThenThen You Have A Right Triangle You Have A Right Triangle
aa
bb
cc
Do These Lengths Form Right Triangles?Do These Lengths Form Right Triangles?i.e. do they work in the Pythagorean Theorem?i.e. do they work in the Pythagorean Theorem?
5, 6, 10 6, 8, 105, 6, 10 6, 8, 10
10² __5² + 6² 10² __5² + 6² 100___25 + 36 100___25 + 36
100100≠ ≠ 61 61 NONO
10²___6² + 8² 10²___6² + 8² 100___36 + 64 100___36 + 64
100 100 = = 100 100 YESYES
Example of the ConverseDetermine whether a
triangle with lengths 7, 11, and 12 form a right triangle.
**The hypotenuse is the longest length.
12149144
11712?
22?
2
170144
This is not a right triangle.
A A Pythagorean Triple Pythagorean Triple Is Any 3 Is Any 3 Integers Integers That Form A Right TriangleThat Form A Right Triangle
3, 4, 53, 4, 5Multiples FamilyMultiples Family
6,8,106,8,1030,40,5030,40,5015,20,2515,20,25
5, 12, 135, 12, 13Multiples FamilyMultiples Family
10,24,2610,24,2625,60,6525,60,6535,84,9135,84,91
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.
256144400
161220?
22?
2
400400This 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.
Determine whether
15, 8, and 17 is a Pythagorean triple.
251636
546?
22?
2
41364, 5, and 6 is not a Pythagorean triple.
64225289
81517?
22?
2
289289
15, 8, and 17 is a Pythagorean triple.
Verifying Right Triangles
78
?? The triangle is
a right triangle.
Note: squaring a square root!!
36
15
Verifying Right Triangles
The triangle is NOT a
right triangle.
???
Note: squaring an integer & square root!!
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 ? What Kind Of Triangle ? c²c² ?? a² + b² ?? a² + b²
If the If the c²c² = = a² + b² , then a² + b² , then rightright
If the If the c²c² >> a² + b² then a² + b² then obtuseobtuse
If the If the c²c² < < a² + b², then a² + b², then acuteacute
What Kind Of Triangle ? What Kind Of Triangle ? c²c² ?? a² + b² ?? a² + b²
The converse of the Pythagorean Theorem can be used to categorize triangles.
Triangle Inequality
Triangle Inequality
38, 77, 86
c2 ? a2 + b2
862 ? 382 + 772
7396 ? 1444 + 5959
7396 > 7373
The triangle is obtuse
Triangle Inequality10.5, 36.5, 37.5
c2 ? a2 + b2
37.52 ? 10.52 + 36.52
1406.25 ? 110.25 + 1332.25
1406.24 < 1442.5
The triangle is acute
4,7,99²__4² + 7² 9²__4² + 7² 81__16 + 49 81__16 + 49 81 > 6581 > 65 OBTUSEOBTUSE
greatergreater
5,5,77² __5² + 5² 7² __5² + 5² 4949__ 25 +25 __ 25 +25
49 < 5049 < 50
ACUTEACUTE
Less thanLess than
259
16
52=32
+ 42
25=9 + 16
A Pythagorean Triple
3
4
5
3, 4, 5
In a right-angled triangle, the square on
the hypotenuse is equal to the
sum of the squares on the
other two sides.
169169
144144
2255 13
2 =5
2 + 12
2
169=25 + 144
A 2nd Pythagorean Triple
5, 12, 13
5
12
13
In a right-angled triangle, the square on
the hypotenuse is equal to the
sum of the squares on the
other two sides.
625
576
49
252 =7
2+
242
625=49 + 576
7
24
25
A 3rd Pythagorean
Triple7, 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 foundationSolution: 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