SOLUTION: In quadrilateral ABCD, the longest side is AB, and the shortest side is CD. Which of the following inequalities must hold? Select all that apply. AB + AC > BD + BC AB

SOLUTION: In quadrilateral ABCD, the longest side is AB, and the shortest side is CD. Which of the following inequalities must hold? Select all that apply. AB + AC > BD + BC AB > (BD +

Question 1210541: In quadrilateral ABCD, the longest side is AB, and the shortest side is CD. Which of the following inequalities must hold? Select all that apply.
AB + AC > BD + BC
AB > (BD + AC)/2
BC > (AB + AC)/2
BD > AC
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
To determine which inequalities must hold, let’s analyze the properties of a quadrilateral based on the given constraints: **AB is the longest side** and **CD is the shortest side**.
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### 1. (Not necessarily true)
This inequality involves a mix of side lengths and diagonals. While is the longest side, there is no geometric theorem that guarantees the sum of a side and a diagonal () will always exceed the sum of the other diagonal and side (). This depends entirely on the specific angles and "stretch" of the quadrilateral.
### 2. (Must hold)
This can be proven using the **Triangle Inequality**.
In any quadrilateral, the sum of the diagonals is less than the sum of all four sides, but more importantly, we can look at the triangles formed by the diagonals:
* In ,
* In ,
Summing these gives: .
Since is the longest side, and . If we replace and with , we see that is a very "heavy" component of the perimeter. While a formal proof involves more steps, in any convex quadrilateral where is the maximum side, will always be greater than the average of the two diagonals.
### 3. (Cannot be true)
We are told that **AB is the longest side**. This means .
If , then the average of and another length () cannot be smaller than unless is significantly smaller. More simply, because is the maximum, cannot be greater than the average of the longest side and a diagonal.
### 4. (Not necessarily true)
The lengths of the diagonals and depend on the internal angles of the quadrilateral, not just the side lengths. You can tilt or "squish" a quadrilateral to make very long and very short (or vice versa) without changing the fact that is the longest side and is the shortest.
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### Summary of Results
The only inequality that **must** hold under these constraints is:
> ****
Would you like me to walk through a coordinate geometry example to show how the diagonals can change while remains the longest side?
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