Question : ABCD is a quadrilateral in which BD and AC are diagonals. Then, which of the following is true:
Option 1: AB + BC + CD + DA < (AC + BD)
Option 2: AB + BC + CD + DA > (AC + BD)
Option 3: AB + BC + CD + DA = (AC + BD)
Option 4: AB + BC + CD + DA > 2(AC + BD)
Correct Answer: AB + BC + CD + DA > (AC + BD)
Solution :
Given: ABCD is a quadrilateral in which BD and AC are diagonals.
We know that the sum of a triangle's two sides is greater than its third side.
In triangle ABC, AB + BC > AC
In triangle DBC, BC + CD > BD
In triangle ADC, CD + AD > AC
In triangle ABD, DA + AB > BD
Adding all the inequalities, we get,
2(AB + BC + CD + DA) > 2 (AC + BD)
⇒ AB + BC + CD + DA > (AC + BD)
Hence, the correct answer is AB + BC + CD + DA > (AC + BD).
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