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Same Perimeter and Area, but Different Triangle

Source: 2015 AMC 12A #20

February 4, 2015
geometryperimeterquadraticsarea of a triangleHeron's formulaalgebraquadratic formula

Problem Statement

Isosceles triangles TT and T′T' are not congruent but have the same area and the same perimeter. The sides of TT have lengths 55, 55, and 88, while those of T′T' have lengths aa, aa, and bb. Which of the following numbers is closest to bb?
<spanclass=′latex−bold′>(A)</span>3<spanclass=′latex−bold′>(B)</span>4<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>6<spanclass=′latex−bold′>(E)</span>8<span class='latex-bold'>(A) </span>3\qquad<span class='latex-bold'>(B) </span>4\qquad<span class='latex-bold'>(C) </span>5\qquad<span class='latex-bold'>(D) </span>6\qquad<span class='latex-bold'>(E) </span>8
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