The shaded region in the figure above represents a frame shaped as a right triangle with sides \(AB = 15\) cm and \(AC = 17\) cm. The frame ABC encloses a picture, shaped also as a right triangle, whose area is \(1/2\) of the area of the frame. What is the length of the hypotenuse of the triangular picture if the length of the other two sides is 14 cm ?

Explanation: Given, \(AB=15\) cm and \(AC=17\) cm. So, \(BC=8\) cm. The area of the triangle ABC is \(0.5 * 15 * 8 = 60\) sq. cm. Now sum of two sides of the picture is 14 . Let the sides be \(x\) and \((14-x)\). So, \(3 * 0.5 * x * (14-x) = 60\). Or, \(x=10, 4\). Two sides of the picture are 4 and 10 . So, using Pythagoras' theorem, the length of the hypotenuse is \(\sqrt{4^2+10^2} = \sqrt{16+100} = \sqrt{116} = 2\sqrt{29}\) cm.