Rhomboid

The rhomboid sides' dimensions are a= |AB|=5cm, b = |BC|=6 cm, and the angle's size at vertex A is 60°. What is the length of the diagonal AC?

Final Answer:

AC =  9.54 cm

Step-by-step explanation:

$$\begin{gathered} a=5\text{cm} \\ b=6\text{cm} \\ \alpha =60{}^{\circ } \\ \beta =180-\alpha =180-60=120{}^{\circ } \\ \cos \alpha =a_1:b \\ \sin \alpha =h:b \\ {a}_1=b\cdot \cos \alpha =b\cdot \cos 60°=6\cdot \cos 60°=6\cdot 0.5=3\text{cm} \\ h=b\cdot \sin \alpha =b\cdot \sin 60°=6\cdot \sin 60°=6\cdot 0.866025=5.19615\text{cm} \\ {a}_2=a+a_1=5+3=8\text{cm} \\ A{C}^2=a_2^2+h^2 \\ AC=\sqrt{a_2^2+h^2}=\sqrt{8^2+5.1962^2}=\sqrt{91}=9.54\text{cm} \end{gathered}$$

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