Diagonals of the rhombus

How long are the diagonals e, and f in the diamond if its side is 5 cm long and its area is 20 cm²?

Correct answer:

e = 4.4721 cm
f = 8.9443 cm

a = 5 cm S = 20 cm^{2} S = a h h = S / a = 20 / 5 = 4 cm h = a \cdot sin (A) A = \frac{1 8 0 °}{π} \cdot arcsin (h / a) = \frac{1 8 0 °}{π} \cdot arcsin (4 / 5) ≐ 53.1301^{\circ} e = a \cdot 2 - 2 \cdot cos A = a \cdot 2 - 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 - 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 - 2 \cdot 0.6 = 4.472 = 4.4721 cm

f = a \cdot 2 + 2 \cdot cos A = a \cdot 2 + 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 + 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 + 2 \cdot 0.6 = 8.94427 = 8.9443 cm Verifying Solution: S = E F / 2 (E / 2)^{2} + (F / 2)^{2} = a^{2} (E / 2)^{2} + (2 \cdot S / E / 2)^{2} = a^{2} E^{2} + 4 \cdot S^{2} / E^{2} = 4 a^{2} z = E^{2} z + 4 \cdot S^{2} / z = 4 a^{2} z^{2} + 4 \cdot S^{2} = 4 \cdot a^{2} \cdot z z^{2} + 4 \cdot 2 0^{2} = 4 \cdot 5^{2} \cdot z z^{2} - 100 z + 1600 = 0 a = 1; b = - 100; c = 1600 D = b^{2} - 4 a c = 10 0^{2} - 4 \cdot 1 \cdot 1600 = 3600 D > 0 z_{1, 2} = \frac{- b \pm D}{2 a} = \frac{1 0 0 \pm 3 6 0 0}{2} z_{1, 2} = \frac{1 0 0 \pm 6 0}{2} z_{1, 2} = 50 \pm 30 z_{1} = 80 z_{2} = 20 z = z_{1} = 80 E = z = 80 = 45 cm ≐ 8.9443 cm F = 2 \cdot S / E = 2 \cdot 20 / 8.9443 = 25 cm ≐ 4.4721 cm

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