Mathematics
In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.
Triangles
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Answer
Given: ABCD is a quadrilateral such that DC is the largest side.
To prove: AB + AD > DC - BC
Construction: Join diagonal AC.
Proof: According to the triangle inequality property, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In Δ ABC,
⇒ AB + BC > AC ……………(1)
Similarly, in Δ ADC,
⇒ AD + AC > DC ……………(2)
Adding equation (1) and (2), we get:
⇒ AB + BC + AD + AC > AC + DC
⇒ AB + BC + AD > DC
⇒ AB + AD > DC - BC
Hence, proved AB + AD > DC - BC.
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