Question

find the length of side b.
32° b
c 7
125°
a b c
a = 23° b = ? c =

Explanation:

Step1: Identify the Law of Sines

We use the Law of Sines, which states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ for a triangle with sides $a, b, c$ and opposite angles $A, B, C$ respectively. Here, side opposite angle $B$ (which is 32°) is $AC = b$? Wait, no, wait. Wait, in triangle $ABC$, angle at $C$ is 125°, angle at $B$ is 32°, angle at $A$ is 23°. The side opposite angle $B$ (32°) is $AC$? Wait, no, side $BC$ is 7. Wait, side $BC$ is opposite angle $A$ (23°), side $AC$ is $b$ (opposite angle $B$ (32°)), and side $AB$ is $c$ (opposite angle $C$ (125°)). Wait, let's clarify:

In triangle $ABC$:

• Angle $A = 23°$, opposite side $BC = 7$ (wait, no, the diagram: point $C$ is between $A$ and $b$, side $BC$ is 7, angle at $C$ is 125°, angle at $B$ is 32°, angle at $A$ is 23°. So side $BC$ is length 7, which is opposite angle $A$ (23°). Side $AC$ is $b$, opposite angle $B$ (32°). Side $AB$ is $c$, opposite angle $C$ (125°).

So by Law of Sines: $\frac{BC}{\sin A}=\frac{AC}{\sin B}=\frac{AB}{\sin C}$

So $BC = 7$, angle $A = 23°$, angle $B = 32°$, angle $C = 125°$, and $AC = b$.

So $\frac{7}{\sin 23°}=\frac{b}{\sin 32°}$

Step2: Solve for $b$

We can rearrange the formula to solve for $b$:

$b=\frac{7 \times \sin 32°}{\sin 23°}$

First, calculate $\sin 32°$ and $\sin 23°$:

$\sin 32° \approx 0.5299$

$\sin 23° \approx 0.3907$

Then:

$b=\frac{7 \times 0.5299}{0.3907}$

Calculate numerator: $7 \times 0.5299 = 3.7093$

Then divide by 0.3907: $3.7093 \div 0.3907 \approx 9.5$ (approximate value, more precise calculation: 3.7093 / 0.3907 ≈ 9.494, so approximately 9.5)

Wait, let's do more precise calculation:

$\sin 32° = \sin(32) \approx 0.5299192642$

$\sin 23° = \sin(23) \approx 0.3907311285$

So $b = \frac{7 \times 0.5299192642}{0.3907311285}$

Calculate numerator: 7 * 0.5299192642 = 3.709434849

Divide by 0.3907311285: 3.709434849 / 0.3907311285 ≈ 9.493, so approximately 9.5 (or more precisely, 9.5 when rounded to one decimal place, or 9.49 if two decimal places)

Answer:

The length of side \( b \) is approximately \(\boxed{9.5}\) (or more precisely, approximately 9.49 depending on rounding).