Geometry: An Interactive Journey to Mastery

Lesson 5: The Pythagorean Theorem

The Pythagorean Theorem

Lesson 5

Topics

x 7KH3\WKDJRUHDQWKHRUHP

x Shortest distances.

x 7KHWULDQJXODULQHTXDOLW\

'H¿QLWLRQ

x hypotenuse7KHVLGHRSSRVLWHWKHULJKWDQJOHLQDULJKWWULDQJOH

Results

x 3\WKDJRUHDQWKHRUHP ,IVTXDUHVDUHGUDZQLQWKHVLGHVRIDULJKWWULDQJOHWKHQWKHDUHDVRIWKHWZR

VPDOOHUVTXDUHVVXPWRWKHDUHDRIWKHODUJHVTXDUHͼ௘$VDVWDWHPHQWRIDOJHEUD)RUDULJKWWULDQJOHZLWK

VLGHVRIOHQJWKpqDQGrZKHUHrLVWKHOHQJWKRIWKHVLGHRSSRVLWHWKHULJKWDQJOHp^2 q^2 = r^2 ௘ͽ

x WULDQJXODULQHTXDOLW\ ,QDWULDQJOHWKHVXPRIDQ\WZRVLGHOHQJWKVRIWKHWULDQJOHLVVWULFWO\JUHDWHU

WKDQWKHOHQJWKRIWKHUHPDLQLQJVLGH

Summary

,QWKLVOHVVRQZHLQWURGXFHWKHIDPRXV3\WKDJRUHDQWKHRUHPDQGGHYHORSDSRVVLEOHSURRIRILW8QFHUWDLQW\

DERXWWKHYDOLGLW\RIWKHSURRIOHDGVXVWRDFFHSWWKH3\WKDJRUHDQWKHRUHPDVDVHFRQGIXQGDPHQWDODVVXPSWLRQ

RIJHRPHWU\$VORJLFDOFRQVHTXHQFHVRIWKHWKHRUHPZHGHPRQVWUDWHWKDWVWUDLJKWSDWKVUHSUHVHQWWKHVKRUWHVW

URXWHVEHWZHHQWZRSRLQWVDQGSURYHWKHWULDQJXODULQHTXDOLW\

Example 1

)LQGWKHOHQJWKRIWKHOLQHVHJPHQWABJLYHQDVWKHLQWHULRU

GLDJRQDORIWKHUHFWDQJXODUER[VKRZQLQFigure 5.1.

Solution

7KHOHQJWKACLVWKHK\SRWHQXVHRIDULJKWWULDQJOHZLWKOHJVDQG7KXVAC^2 = 3^2 ^2  JLYLQJAC 

7KHOHQJWKABWKHOHQJWKZHVHHNLVWKHK\SRWHQXVHRIDULJKWWULDQJOHZLWKOHJVAC DQG

7KXVAB^2  ^2 ^2  JLYLQJAB 

C

A

B

3 4

12

Figure 5.1