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# For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

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For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

Nov 14, 2020

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Using the triangle inequality which says that the sum of any two sides of a triangle is greater than the remaining side, we have two triangles to consider

Top triangle....

22 + 18   = 40

So the greatest integer value that we can have for the dashed line is 39

18 + x  must be greater than 22

So the least integer value that we can have for the dashed line is 5

So the interval  that  solves  the dashed line in the top riangle is  [ 5, 39]     (1)

Bottom triangle

21 + 25 =  46

So the greatest integer value that we can have for the dashed line is  45

21 + x  = must be greater than 25

So the least integer value that we can have for the dashed line is 5

So the interval that solves the dashed line in the bottom triangle is  [5, 45 ]    (2)

Taking the intersection of (1)  and (2)  we have  that the   interval that solves the problem is

[ 5, 39 ]

So....the number of whole numbers that satisfies  the dashed line length  is

39  - 5  +  1   =

34  +  1  =

35

Nov 14, 2020