Respuesta :

luisejr77

Answer:

[tex]c=20.6[/tex]

Step-by-step explanation:

We know that  m∠A=23°, a=10, and b=13

To find c we must first find B and C. Then we use the sine theorem

sine theorem:

[tex]\frac{sin(A)}{a}=\frac{sin(B)}{b} =\frac{sin(C)}{c}[/tex]

then:

[tex]\frac{sin(A)}{a}=\frac{sin(B)}{b}[/tex]

[tex]\frac{sin(23)}{10}=\frac{sin(B)}{13}[/tex]

[tex]0.0391=\frac{sin(B)}{13}[/tex]

[tex]Arcsin(13*0.0391)=B[/tex]

[tex]B=30.55[/tex]

So

We know that:

[tex]23+30.55+C=180\\C=180-23-30.55\\C=126.45[/tex]

we use the sine theorem again to find the length c

[tex]\frac{sin(23)}{10}=\frac{sin(C)}{c}[/tex]

[tex]0.0391=\frac{sin(C)}{c}[/tex]

[tex]c=\frac{sin(126.45)}{0.0391}[/tex]

[tex]c=20.6[/tex]

Answer:

Using Sine Rule

[tex]\rightarrow \frac{a}{\sinA}=\frac{b}{\sin B}=\frac{c}{\sin C}\\\\ \rightarrow \frac{10}{\sin 23^{\circ}}=\frac{13}{\sin B}\\\\ \sinB=\frac{13 \times 0.39}{10}\\\\ \sin B=\frac{5.07}{10}\\\\ \sinB=0.507\\\\B=31^{\circ}[/tex]

Using Angle sum property of Triangle

 ∠A+∠B+∠C=180°

23°+31°+∠C=180°

∠C=180°-54°

∠C=126°

Again using Sine rule

[tex]\frac{c}{\sin C}=\frac{b}{\sin B}\\\\ \frac{c}{\sin126^{\circ}}=\frac{13}{0.507}\\\\c=\frac{0.809 \times 13}{0.507}\\\\c=\frac{10.517}{0.507}\\\\c=20.7435[/tex]

Length of third side=20.75(approx)

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