Diamond diagonals

Find the diamond diagonal's lengths if the area is 156 cm² and the side is 13 cm long.

Correct answer:

u = 21.6333 cm
v = 14.4222 cm

S = 156 cm^{2} a = 13 cm S = a h = \frac{u v}{2} h = S / a = 156 / 13 = 12 cm (u / 2)^{2} + (v / 2)^{2} = a^{2} u^{2} + v^{2} = 4 a^{2} u^{2} + v^{2} = 1352 u v = 2 S = 312 u^{2} + (2 S / u)^{2} = 4 a^{2} u^{2} + 31 2^{2} / u^{2} = 676 x = u^{2} x + 97344 / x = 676 x^{2} + 97344 = 676 x x^{2} - 676 x + 97344 = 0 a = 1; b = - 676; c = 97344 D = b^{2} - 4 a c = 67 6^{2} - 4 \cdot 1 \cdot 97344 = 67600 D > 0 x_{1, 2} = \frac{- b \pm D}{2 a} = \frac{6 7 6 \pm 6 7 6 0 0}{2} x_{1, 2} = \frac{6 7 6 \pm 2 6 0}{2} x_{1, 2} = 338 \pm 130 x_{1} = 468 x_{2} = 208 u_{1} = x_{1} = 468 = 613 ≐ 21.6333 u_{2} = x_{2} = 208 = 413 ≐ 14.4222 u = u_{1} = 21.6333 = 613 = 21.6333 cm

v = 2 \cdot S / u = 2 \cdot 156 / 21.6333 = 413 ≐ 14.4222 cm v = u_{2} S_{2} = \frac{u \cdot v}{2} = \frac{2 1 . 6 3 3 3 \cdot 1 4 . 4 2 2 2}{2} = 156 cm^{2} a_{2} = (u / 2)^{2} + (v / 2)^{2} = (21.6333 / 2)^{2} + (14.4222 / 2)^{2} = 13 cm

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