Question

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?

Answer 1

Answer

The approximate values are:

c = 55.2°

r = 22.8°

x = 9.9 miles

Explanation

- To find angle $c$, we are using the rule of sines: $\frac{a}{sin(A)} =\frac{b}{sin(B)} =\frac{c}{sin(C)}$

For our triangle $a=21,A=c,b=x,B=r,c=25$ and $C=102$

Replacing the values we get: $\frac{21}{sin(c)} =\frac{x}{sin(r)} =\frac{25}{sin(102)}$

We can pick up two suited values to find $c$:

$\frac{21}{sin(c)} =\frac{25}{sin(102)}$

$21=\frac{25sin(c)}{sin(102)}$

$21sin(102)=25sin(c)$

$sin(c)=\frac{21sin(102)}{25}$

$c=sin^{-1}(\frac{21sin(102)}{25})$

$c=55.2$

- Now that we have angle $c$, we can use the angle sum theorem to find angle $r$.

The angle sum theorem states the the interior angles of a triangle add up to 180°, so:

$r+c+102=180$

$r+55.2+102=180$

$r+157.2=180$

$r=22.8$

- Now that we have angle $r$, we can use the rule of sines, one more time, to find side $x$

$\frac{21}{sin(c)} =\frac{x}{sin(r)} =\frac{25}{sin(102)}$

$\frac{x}{sin(r)} =\frac{25}{sin(102)}$

$\frac{x}{sin(22.8)} =\frac{25}{sin(102)}$

$x=\frac{25sin(22.8)}{sin(102)}$

$x=9.9$

Answer 2

Answer: 1) r=23° , c=55° , x=10°

Step-by-step explanation:

correct on edge 2020