SOLUTION: Two side of a parallelogram are 694 feet and 418 feet respectively, one diagonal is 602 feet. Find the length of the other diagonal.

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Question 1019495: Two side of a parallelogram are 694 feet and 418 feet
respectively, one diagonal is 602 feet. Find the length
of the other diagonal.

Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!




We will be using the law of cosines.
Since the adjacent interior angles of a parallogram are
supplementary, cos(∠ABC) = -cos(∠DAB), so

Using the law of cosines on ΔABD

       BD² =  AD² +  AB² - 2∙ AD∙ AB∙cos(∠DAB)
(1)   602² = 418² + 694² - 2∙418∙694∙cos(∠DAB)  

Using the law of cosines on ΔABC

       AC² =  BC² +  AB² - 2∙ BC ∙AB∙cos(∠ABC)
       AC² = 418² + 694² - 2∙418∙694∙[-cos(∠DAB)]
(2)    AC² = 418² + 694² + 2∙418∙694∙cos(∠DAB)

Adding equations (1) and (2)

(1)   602² = 418² + 694² - 2∙418∙694∙cos(∠DAB)
(2)    AC² = 418² + 694² + 2∙418∙694∙cos(∠DAB)

602² + AC² = 2∙418² + 2∙694² 
       AC² = 2∙418² + 2∙694² - 602²
       AC² = 2∙418² + 2∙694² - 602²
       AC² = 349448 + 963272 - 362484
       AC² = 950316
        AC = √950316
        AC = 974.8415256 ft.   

Edwin

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Two side of a parallelogram are 694 feet and 418 feet
respectively, one diagonal is 602 feet. Find the length
of the other diagonal.
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You can get the solution quickly, if you apply this property of a parallelogram diagonals:


      Let  a,  b,  c and  d be the lengths of the sides of a parallelogram and    and    be the lengths of its diagonals.
      Then      = = .


It is proved in the lesson  The length of diagonals of a parallelogram  in this site.

When you substitute your data,  you will have

= ,

Then you get    = = 950316  and    = = 974.8415 ft.


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