Question 556147: ABCD is a quadrilateral with AB=8, BC=13, and DC=15. What is the maximum possible integral value of AD?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe the minimum integral value would be 11 and the maximum integral value would be 14.
here's why:
assume the left side of the quadrilateral is perpendicular to the top and bottom sides of the quadrilateral.
that's the shortest it can be.
now assume the right side of the quadrilateral is perpendicular to the top and bottom sides of the quadrilateral.
that's the shortest the right side can be which results in the longest the left side can be.
you would get a figure that looks like the following:
the top figure is the shortest that the left side of the quadrilateral can be.
the bottom figure is the longest that the left side of the quadrilateral can be.
Answer by ikleyn(52814)  (Show Source):
You can put this solution on YOUR website! .
ABCD is a quadrilateral with AB=8, BC=13, and DC=15. What is the maximum possible integral value of AD?
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The logic and the answer in the post by tutor @Theo are incorrect.
A came to bring a correct solution.
Obviously, the length of AD is always less than the sum AB + BC + CD = 8 + 13 + 15 = 36.
This is true due to the axiom of Geometry, which states that a straight line segment
connecting two points on a plane is always shorter than any polyline connecting these points.
In principle, the length of AD can be as close as desired to the sum of the three sides,
but never reaches the value of the sum (until the quadrilateral is not degenerated).
So, the length of AD has no maximum in the strict mathematical sense, although it is bounded
from the top by the value of the sum, 36.
But since the problem asks about the maximum INTEGRAL value of AD,
this maximum integral length of AD do exist, and is equal to 35 units.
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