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

a b c

The Pythagorean Theorem as some students see it.

c a b c 2 = a 2 + b 2

A better way

a

2 a c

c

2 b

b

2

c

2

=a

2

+b

2

PYTHAGOREAN THEOREM

Applies to Right Triangles Only!

leg a c hypotenuse b leg c 2 =a 2 +b 2

2 5 cm Pythagoras Questions 1 3 cm x 4 cm

Pythagorean triple

x 12 cm

Pythagorean triple

4 11 cm Pythagoras Questions: Finding a leg measure 3 x m 11m 9 m Another method for finding a leg measure

x ≈ 6.32 cm

23.8 cm x cm

x ≈ 21.11 cm

1 Applications of Pythagoras Find the diagonal of the rectangle 6 cm d 9.3 cm

d = 11.07 cm

2 4.3 cm A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm. Find its perimeter.

7.8 cm

x

cm

therefore

x ≈ 6.51 cm



Perimeter = 2(6.51+4.3) ≈ 21.62 cm

The Converse Of The Pythagorean Theorem

If c² =a² + b² , Then You Have A Right Triangle a c 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 ≠ 61 100 = 100 NO YES

Example of the Converse Determine whether a triangle with lengths 7, 11, and 12 form a right triangle.

**The hypotenuse is the longest length.

12 2 ?

 7 2  11 2 144 ?

 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

Multiples Family 6,8,10 30,40,50 15,20,25

5, 12, 13

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 2 ?

 12 2  16 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 36 ?

 4 2  41  5 2 36 ?

 16  25 Determine whether 15, 8, and 17 is a Pythagorean triple.

17 2 ?

 15 2  8 2 289 ?

 289  225  289 64

15, 8, and 17 is a Pythagorean triple .

Verifying Right Triangles 8 7

?

?

Note: squaring a square root!!

The triangle is a right triangle.

Verifying Right Triangles

?

?

?

36

Note: squaring an integer & square root!!

The triangle is NOT a right triangle.

15

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 t riangles.

Triangle Inequality

38, 77, 86

c 2 ? a 2 + b 2 86 2 ? 38 2 + 77 2 7396 ? 1444 + 5959 7396 > 7373 The triangle is obtuse

Triangle Inequality

10.5, 36.5, 37.5

c 2 ? a 2 + b 2 37.5

2 ? 10.5

2 + 36.5

2 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 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares the other two sides.

on 9 3 5 25 4 16 5 2 =3 2 + 4 2

25=9 + 16

A 2nd Pythagorean Triple 5, 12, 13 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares the other two sides.

on 25 5 169 13 12 144 13 2 =5 2 + 12 2 169=25 + 144

A 3 rd Pythagorean Triple 625 49 7 25 24 25 2 =7 2 + 24 2 625=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