Answer:
[tex]\sf A \approx 27^\circ [/tex]
[tex]\sf a \approx 5.1 [/tex]
[tex]\sf c \approx 11.1 [/tex]
Step-by-step explanation:
To use the Law of Sines, we have the following relationship:
[tex] \large\boxed{\boxed{\sf \dfrac{\sin(A)}{a} = \dfrac{\sin(B)}{b} = \dfrac{\sin(C)}{c}}} [/tex]
Given:
- [tex]\sf B = 53^\circ [/tex]
- [tex]\sf C = 100^\circ [/tex]
- [tex]\sf b = AC = 9 [/tex]
To find [tex]\sf A [/tex], we'll use the relationship between angles and sides. We know that the sum of angles in a triangle is [tex]\sf 180^\circ [/tex], so we can find [tex]\sf A [/tex] as follows:
[tex]\sf A = 180^\circ - B - C [/tex]
[tex]\sf A = 180^\circ - 53^\circ - 100^\circ [/tex]
[tex]\sf A = 27^\circ [/tex]
Now, to find [tex]\sf a [/tex], we'll use the Law of Sines:
[tex]\sf \dfrac{\sin(A)}{a} = \dfrac{\sin(B)}{b} [/tex]
[tex]\sf \dfrac{\sin(27^\circ)}{a} = \dfrac{\sin(53^\circ)}{9} [/tex]
[tex]\sf a = \dfrac{9 \times \sin(27^\circ)}{\sin(53^\circ)} [/tex]
Using a calculator, we find:
[tex]\sf a \approx \dfrac{9 \times 0.4539904997395}{0.7986355100472} [/tex]
[tex]\sf a \approx \dfrac{4.086}{0.798} [/tex]
[tex]\sf a \approx 5.1161192386924 [/tex]
[tex]\sf a \approx 5.1 \textsf{(in nearest tenth)}[/tex]
To find [tex]\sf c [/tex], we'll use the Law of Sines again:
[tex]\sf \dfrac{\sin(C)}{c} = \dfrac{\sin(B)}{b} [/tex]
[tex]\sf \dfrac{\sin(100^\circ)}{c} = \dfrac{\sin(53^\circ)}{9} [/tex]
[tex]\sf c = \dfrac{9 \times \sin(100^\circ)}{\sin(53^\circ)} [/tex]
Using a calculator, we find:
[tex]\sf c \approx \dfrac{9 \times 0.9848077530122}{0.7986355100472} [/tex]
[tex]\sf c \approx 11.098016135777[/tex]
[tex]\sf c \approx 11.1 \textsf{(in nearest tenth)}[/tex]
So, the missing measures of ∆ ABC are:
[tex]\sf A \approx 27^\circ [/tex]
[tex]\sf a \approx 5.1 [/tex]
[tex]\sf c \approx 11.1 [/tex]
