Rhombus

A rhombus has a side length of a = 20 cm. The points where the inscribed circle touches the sides divide each side into segments of length a₁ = 13 cm and a₂ = 7 cm. Calculate the radius r of the inscribed circle and the lengths of the diagonals of the rhombus.

Final Answer:

r =  9.5 cm
u₁ =  32.2 cm
u₂ =  23.7 cm

Step-by-step explanation:

$$\begin{gathered} a_1=13\text{cm} \\ {a}_2=7\text{cm} \\ a=a_1+a_2=13+7=20\text{cm} \\ {r}^2=a_1\cdot a_2 \\ r=\sqrt{a_1\cdot a_2}=\sqrt{13\cdot 7}=\sqrt{91}=9.5\text{cm} \end{gathered}$$

$$\begin{gathered} (u_1/2{)}^2=r^2+a_1^2 \\ {u}_1=2\cdot \sqrt{r^2+a_1^2}=2\cdot \sqrt{9.5394^2+13^2}=4\sqrt{65}=32.2\text{cm} \end{gathered}$$

$$\begin{gathered} (u_2/2{)}^2=r^2+a_2^2 \\ {u}_2=2\cdot \sqrt{r^2+a_2^2}=2\cdot \sqrt{9.5394^2+7^2}=4\sqrt{35}=23.7\text{cm} \end{gathered}$$

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