Is there a 4-gon with given side lengths? (i) Recall the following fact about triangles and check it by construction. For given positive numbers a, b, c, is there a triangle whose sides have these lengths? The answer is Yes exactly when the sum of any two numbers is greater than the third. If we arrange the numbers in increasing order (suppose a ≤ b ≤ c), then this amounts to requiring a + b > c. (Hint: Start by drawing a segment of length c.) (ii) Suppose a 4-gon has 2, 5, 11 as three side lengths. Can the length of the fourth side be 100? Can it be 10? Can it be 1? What are the possible lengths of the fourth side? (iii) For given positive numbers a, b, c, d, how will you decide if there is a 4-gon whose sides have these lengths?
(ii) A polygon can close only when its longest side is strictly less than the sum of all remaining sides. For side lengths 2, 5, 11 and fourth side x, we must have x < 2 + 5 + 11 = 18 and 11 < 2 + 5 + x, which gives 4 < x < 18. Hence, 100 is impossible, 1 is impossible and 10 is possible. (iii) A 4-gon exists if and only if each side is strictly smaller than the sum of the other three sides.
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