Determine 83081

A paper kite is shaped like a deltoid ABCD, with two shorter sides 30 cm long, two longer sides 51 cm long, and a shorter diagonal 48 cm long. Determine the sizes of the internal angles of the given deltoid.

Correct answer:

α =  106.2602 °
β =  98.7974 °
γ =  56.145 °
δ =  98.7974 °

Step-by-step explanation:

$$\begin{gathered} a=30\text{cm} \\ b=51\text{cm} \\ u=48\text{cm} \\ s=u/2=48/2=24\text{cm} \\ \sin A=s:a \\ A=\frac{180°}{\pi }\cdot \arcsin (s/a)=\frac{180°}{\pi }\cdot \arcsin (24/30)\doteq 53.1301{}^{\circ } \\ \sin B=s:b \\ B=\frac{180°}{\pi }\cdot \arcsin (s/b)=\frac{180°}{\pi }\cdot \arcsin (24/51)\doteq 28.0725{}^{\circ } \\ {A}_2=90-A=90-53.1301\doteq 36.8699{}^{\circ } \\ {B}_2=90-B=90-28.0725\doteq 61.9275{}^{\circ } \\ \alpha =2\cdot A=2\cdot 53.1301=106.2602^{\circ }=106°15^{\prime }37" \end{gathered}$$

$\beta =A_2+B_2=36.8699+61.9275=98.7974^{\circ }=98°47^{\prime }51"$

$\gamma =2\cdot B=2\cdot 28.0725=56.145^{\circ }=56°8^{\prime }42"$

$\delta =A_2+B_2=36.8699+61.9275=98.7974^{\circ }=98°47^{\prime }51"$

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