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10.5 – The Pythagorean Theorem10.5 – The Pythagorean Theorem
legleg
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hypotenusehypotenuse
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hypotenusehypotenuse
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Pythagorean TheoremPythagorean Theorem: For a right triangle with legs : For a right triangle with legs aa and and bb and hypotenuse and hypotenuse cc, ,
then then aa22 + + bb22 = = cc22, where , where aa is the is the smaller of the two legs.smaller of the two legs.
Ex. 1Ex. 1 Find the length of the missing side. Find the length of the missing side.
a 25a 25
1010
Ex. 1Ex. 1 Find the length of the missing side. Find the length of the missing side.
a 25a 25
1010
aa22 + + bb22 = = cc22
Ex. 1Ex. 1 Find the length of the missing side. Find the length of the missing side.
a 25a 25
1010
aa22 + + bb22 = = cc22
aa22 + (10) + (10)22 = (25) = (25)22
Ex. 1Ex. 1 Find the length of the missing side. Find the length of the missing side.
a 25a 25
1010
aa22 + + bb22 = = cc22
aa22 + (10) + (10)22 = (25) = (25)22
aa22 + 100 = 625 + 100 = 625
Ex. 1Ex. 1 Find the length of the missing side. Find the length of the missing side.
a 25a 25
1010
aa22 + + bb22 = = cc22
aa22 + (10) + (10)22 = (25) = (25)22
aa22 + 100 = 625 + 100 = 625
aa22 = 525 = 525
Ex. 1Ex. 1 Find the length of the missing side. Find the length of the missing side.
a 25a 25
1010
aa22 + + bb22 = = cc22
aa22 + (10) + (10)22 = (25) = (25)22
aa22 + 100 = 625 + 100 = 625
aa22 = 525 = 525
aa ≈ 22.91 ≈ 22.91
Ex. 2Ex. 2 Determine whether the following side Determine whether the following side measures form right triangles.measures form right triangles.
a. 20, 21, 29a. 20, 21, 29
Ex. 2Ex. 2 Determine whether the following side Determine whether the following side measures form right triangles.measures form right triangles.
a. 20, 21, 29a. 20, 21, 29
Shorter lengths Shorter lengths mustmust be the be the legs. legs.
Ex. 2Ex. 2 Determine whether the following side Determine whether the following side measures form right triangles.measures form right triangles.
a. 20, 21, 29a. 20, 21, 29
Shorter lengths Shorter lengths mustmust be the be the legs. legs.
So, So, aa22 + + bb22 = = cc22
Ex. 2Ex. 2 Determine whether the following side Determine whether the following side measures form right triangles.measures form right triangles.
a. 20, 21, 29a. 20, 21, 29
Shorter lengths Shorter lengths mustmust be the be the legs. legs.
So, So, aa22 + + bb22 = = cc22
(20)(20)22 + (21) + (21)22 = (29) = (29)22
Ex. 2Ex. 2 Determine whether the following side Determine whether the following side measures form right triangles.measures form right triangles.
a. 20, 21, 29a. 20, 21, 29
Shorter lengths Shorter lengths mustmust be the be the legs. legs.
So, So, aa22 + + bb22 = = cc22
(20)(20)22 + (21) + (21)22 = (29) = (29)22
400 + 441 = 841400 + 441 = 841
Ex. 2Ex. 2 Determine whether the following side Determine whether the following side measures form right triangles.measures form right triangles.
a. 20, 21, 29a. 20, 21, 29
Shorter lengths Shorter lengths mustmust be the be the legs. legs.
So, So, aa22 + + bb22 = = cc22
(20)(20)22 + (21) + (21)22 = (29) = (29)22
400 + 441 = 841400 + 441 = 841
841 = 841 841 = 841
Ex. 2Ex. 2 Determine whether the following side Determine whether the following side measures form right triangles.measures form right triangles.
a. 20, 21, 29a. 20, 21, 29
Shorter lengths Shorter lengths mustmust be the legs. be the legs.
So, So, aa22 + + bb22 = = cc22
(20)(20)22 + (21) + (21)22 = (29) = (29)22
400 + 441 = 841400 + 441 = 841
841 = 841 841 = 841
, so they form a rt. triangle!, so they form a rt. triangle!
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