On a map, the North Carolina cities of Raleigh, Durham, and Chapel Hill form a triangle, as shown below. What are the approximate values of the missing measures on the map?

AnswerThe approximate values are:c = 55.2°r = 22.8°x = 9.9 miles Explanation- To find angle [tex]c[/tex], we are using the rule of sines: [tex]\frac{a}{sin(A)} =\frac{b}{sin(B)} =\frac{c}{sin(C)}[/tex]For our triangle [tex]a=21,A=c,b=x,B=r,c=25[/tex] and [tex]C=102[/tex]Replacing the values we get: [tex]\frac{21}{sin(c)} =\frac{x}{sin(r)} =\frac{25}{sin(102)}[/tex]We can pick up two suited values to find [tex]c[/tex]:[tex]\frac{21}{sin(c)} =\frac{25}{sin(102)}[/tex][tex]21=\frac{25sin(c)}{sin(102)}[/tex][tex]21sin(102)=25sin(c)[/tex][tex]sin(c)=\frac{21sin(102)}{25}[/tex][tex]c=sin^{-1}(\frac{21sin(102)}{25})[/tex][tex]c=55.2[/tex]- Now that we have angle [tex]c[/tex], we can use the angle sum theorem to find angle [tex]r[/tex].The angle sum theorem states the the interior angles of a triangle add up to 180°, so:[tex]r+c+102=180[/tex][tex]r+55.2+102=180[/tex][tex]r+157.2=180[/tex][tex]r=22.8[/tex]- Now that we have angle [tex]r[/tex], we can use the rule of sines, one more time, to find side [tex]x[/tex][tex]\frac{21}{sin(c)} =\frac{x}{sin(r)} =\frac{25}{sin(102)}[/tex][tex]\frac{x}{sin(r)} =\frac{25}{sin(102)}[/tex][tex]\frac{x}{sin(22.8)} =\frac{25}{sin(102)}[/tex][tex]x=\frac{25sin(22.8)}{sin(102)}[/tex][tex]x=9.9[/tex]