Answer:
Part a) [tex]AC=135.95\ m[/tex]
Part b) [tex]BC=124.14\ m[/tex]
The diagram in the attached figure
Step-by-step explanation:
step 1
Find the measure of angle C
we know that
The sum of the interior angles in any triangle must be equal to 180 degrees
so
[tex]m\angle A+m\angle B+m\angle C=180^o[/tex]
substitute the given values
[tex]53^o+61^o+m\angle C=180^o[/tex]
[tex]114^o+m\angle C=180^o[/tex]
[tex]m\angle C=180^o-114^o[/tex]
[tex]m\angle C=66^o[/tex]
step 2
Find the distance AC
Applying the law of sines
[tex]\frac{AB}{sin(C)}=\frac{AC}{sin(B)}[/tex]
see the attached figure to better understand the problem
substitute the given values
[tex]\frac{142}{sin(66^o)}=\frac{AC}{sin(61^o)}[/tex]
[tex]AC=\frac{142}{sin(66^o)}(sin(61^o))[/tex]
[tex]AC=135.95\ m[/tex] ---> rounded to the nearest hundredth
step 3
Find the distance BC
Applying the law of sines
[tex]\frac{AB}{sin(C)}=\frac{BC}{sin(A)}[/tex]
substitute the given values
[tex]\frac{142}{sin(66^o)}=\frac{BC}{sin(53^o)}[/tex]
[tex]BC=\frac{142}{sin(66^o)}(sin(53^o))[/tex]
[tex]BC=124.14\ m[/tex] ---> rounded to the nearest hundredth
