Diamond diagonals

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

Result

u =  21.633 cm
v =  14.422 cm

Solution:

S=156 cm2 a=13 cm  S=ah=uv2  h=S/a=156/13=12 cm  (u/2)2+(v/2)2=a2  u2+v2=4a2 u2+v2=1352  uv=2S=312  u2+(2S/u)2=4a2 u2+3122/u2=676  x=u2  x+97344/x=676  x2676x+97344=0  a=1;b=676;c=97344 D=b24ac=67624197344=67600 D>0  x1,2=b±D2a=676±676002 x1,2=676±2602 x1,2=338±130 x1=468 x2=208   Factored form of the equation:  (x468)(x208)=0 u1=x1=468=6 1321.6333 u2=x2=208=4 1314.4222  u=u1=21.6333=6 1321.6333=21.633  cm S = 156 \ cm^2 \ \\ a = 13 \ cm \ \\ \ \\ S = ah = \dfrac{ uv }{ 2 } \ \\ \ \\ h = S/a = 156/13 = 12 \ cm \ \\ \ \\ (u/2)^2+(v/2)^2 = a^2 \ \\ \ \\ u^2+v^2 = 4a^2 \ \\ u^2+v^2 = 1352 \ \\ \ \\ uv = 2S = 312 \ \\ \ \\ u^2 + (2S/u)^2 = 4a^2 \ \\ u^2 + 312^2/u^2 = 676 \ \\ \ \\ x = u^2 \ \\ \ \\ x + 97344/x = 676 \ \\ \ \\ x^2 -676x +97344 = 0 \ \\ \ \\ a = 1; b = -676; c = 97344 \ \\ D = b^2 - 4ac = 676^2 - 4\cdot 1 \cdot 97344 = 67600 \ \\ D>0 \ \\ \ \\ x_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 676 \pm \sqrt{ 67600 } }{ 2 } \ \\ x_{1,2} = \dfrac{ 676 \pm 260 }{ 2 } \ \\ x_{1,2} = 338 \pm 130 \ \\ x_{1} = 468 \ \\ x_{2} = 208 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -468) (x -208) = 0 \ \\ u_{ 1 } = \sqrt{ x_{ 1 } } = \sqrt{ 468 } = 6 \ \sqrt{ 13 } \doteq 21.6333 \ \\ u_{ 2 } = \sqrt{ x_{ 2 } } = \sqrt{ 208 } = 4 \ \sqrt{ 13 } \doteq 14.4222 \ \\ \ \\ u = u_{ 1 } = 21.6333 = 6 \ \sqrt{ 13 } \doteq 21.6333 = 21.633 \ \text{ cm }

Checkout calculation with our calculator of quadratic equations.

v=2 S/u=2 156/21.633314.4224=14.422  cm   v=u2  S2=u v2=21.6333 14.42242=156 cm2 a2=(u/2)2+(v/2)2=(21.6333/2)2+(14.4224/2)212.9999 cmv = 2 \cdot \ S/u = 2 \cdot \ 156/21.6333 \doteq 14.4224= 14.422 \ \text{ cm } \ \\ \ \\ v = u_{ 2 } \ \\ \ \\ S_{ 2 } = \dfrac{ u \cdot \ v }{ 2 } = \dfrac{ 21.6333 \cdot \ 14.4224 }{ 2 } = 156 \ cm^2 \ \\ a_{ 2 } = \sqrt{ (u/2)^2+(v/2)^2 } = \sqrt{ (21.6333/2)^2+(14.4224/2)^2 } \doteq 12.9999 \ cm



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