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