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.

Correct result:

r =  8.4 cm
u₁ =  19.5 cm
u₂ =  32.6 cm

Solution:

$$\begin{gathered} a_1=5\text{ cm } \\ {a}_2=14\text{ cm } \\ a=a_1+a_2=5+14=19\text{ cm } \\ {r}^2=a_1\cdot a_2 \\ r=\sqrt{a_1\cdot a_2}=\sqrt{5\cdot 14}=\sqrt{70}=8.4\text{ cm} \end{gathered}$$

$$\begin{gathered} (u_1\mathrm{/}2{)}^2=r^2+a_1^2 \\ {u}_1=2\cdot \sqrt{r^2+a_1^2}=2\cdot \sqrt{8.3666^2+5^2}=2\sqrt{95}=19.5\text{ cm} \end{gathered}$$

$$\begin{gathered} (u_2\mathrm{/}2{)}^2=r^2+a_2^2 \\ {u}_2=2\cdot \sqrt{r^2+a_2^2}=2\cdot \sqrt{8.3666^2+14^2}=2\sqrt{266}=32.6\text{ cm} \end{gathered}$$

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