Respuesta :
Answer:
If the area of the triangle ABC is [tex]450 \mathrm{cm}^{2}[/tex] ,then value of a is 27cm.
Solution:
Given Data:
Area of the triangle ABC is [tex]450 \mathrm{cm}^{2}[/tex]
[tex]\angle \mathrm{B}=82^{\circ}[/tex]
[tex]\angle \mathrm{C}=56^{0}[/tex]
To Find:
Value of a?
Step 1:
[tex]\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}[/tex]
[tex]\angle \mathrm{A}=42^{0}[/tex]
Step 2:
[tex]\mathrm{E}=\frac{1}{2} \alpha \mathrm{b} \sin \mathrm{C}[/tex]
By Law of Sines
[tex]\frac{\alpha}{\sin A}=\frac{b}{\sin B}[/tex]
Simplify the above expression
[tex]\mathrm{b}=\frac{\alpha \sin B}{\sin A}[/tex]
[tex]E=\frac{1}{2} \alpha\left(\frac{\alpha \sin B}{\sin A}\right) \sin C[/tex] --- eqn 1
Step 3:
[tex]\begin{aligned} E &=\frac{1}{2} \alpha^{2}\left(\frac{\sin B \sin C}{\sin A}\right) \\ \alpha &=\sqrt{\frac{2 E \sin A}{\sin B \sin C}} \end{aligned}[/tex]
Step 4:
Substitute the A, B and C value from the given Data.
[tex]\alpha=\sqrt{\frac{2.450 \sin 42^{\circ}}{\sin 82^{\circ} \sin 56^{\circ}}}[/tex]
Apply the Sin theta respective value.
[tex]=\sqrt{\frac{602.22}{0,82}}[/tex]
[tex]\alpha=>27 \mathrm{cm}[/tex]
From the above we finally got the value of "a" which is 27 cm.
