Rhombus

It is given a rhombus of side length a = 19 cm. Touch points of inscribed circle divided his sides into sections a₁ = 5 cm and a₂ = 14 cm. Calculate the radius r of the circle and the length of the diagonals of the rhombus.

Result

r =  8.4 cm
u₁ =  19.6 cm
u₂ =  32.7 cm

Solution:

$a_{1}=5 \ \text{cm} \ \\ a_{2}=14 \ \text{cm} \ \\ \ \\ a=a_{1}+a_{2}=5+14=19 \ \text{cm} \ \\ \ \\ r^2=a_{1} \cdot \ a_{2} \ \\ \ \\ r=\sqrt{ a_{1} \cdot \ a_{2} }=\sqrt{ 5 \cdot \ 14 } \doteq \sqrt{ 70 } \doteq 8.3666 \doteq 8.4 \ \text{cm}$

$(u_{1}/2)^2=r^2+a_{1}^2 \ \\ \ \\ u_{1}=2 \cdot \ \sqrt{ r^2+a_{1}^2 }=2 \cdot \ \sqrt{ 8.3666^2+5^2 } \doteq 19.551 \doteq 19.6 \ \text{cm}$

$(u_{2}/2)^2=r^2+a_{2}^2 \ \\ \ \\ u_{2}=2 \cdot \ \sqrt{ r^2+a_{2}^2 }=2 \cdot \ \sqrt{ 8.3666^2+14^2 } \doteq 32.6533 \doteq 32.7 \ \text{cm}$

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