Question

find the length of side c.
32° b
c
7
125°
a b c
a = 23° b = 9.5 c = ?

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in any triangle, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. We know angle $A = 23^\circ$, side $b = 9.5$, angle $B = 32^\circ$, and we need to find side $c$. First, we can use the Law of Sines to set up the proportion for $c$ and $b$.

Step2: Set up the proportion from Law of Sines

From $\frac{c}{\sin C} = \frac{b}{\sin B}$, we can solve for $c$: $c = \frac{b \cdot \sin C}{\sin B}$. We know $b = 9.5$, $\sin B = \sin 32^\circ$, and we need to find $\sin C$. First, find angle $C$? Wait, no, wait, the triangle has angles $A = 23^\circ$, $B = 32^\circ$, so angle $C = 180^\circ - 23^\circ - 32^\circ = 125^\circ$, which matches the diagram. So $\sin C = \sin 125^\circ$.

Step3: Calculate the values

First, find $\sin 32^\circ \approx 0.5299$, $\sin 125^\circ \approx \sin(55^\circ) \approx 0.8192$ (since $\sin(180^\circ - x) = \sin x$). Then plug into the formula: $c = \frac{9.5 \cdot \sin 125^\circ}{\sin 32^\circ}$.

Step4: Compute the numerator and denominator

Numerator: $9.5 \times 0.8192 \approx 7.7824$
Denominator: $0.5299$
Then $c \approx \frac{7.7824}{0.5299} \approx 14.7$ (rounded to a reasonable decimal place, maybe one decimal or as needed. Wait, let's check the calculation more accurately. $\sin 32^\circ \approx 0.5299192642$, $\sin 125^\circ = \sin(55^\circ) \approx 0.8191520443$. So $9.5 \times 0.8191520443 = 9.5 \times 0.8191520443 \approx 7.781944421$. Then divide by $0.5299192642$: $7.781944421 \div 0.5299192642 \approx 14.7$ (wait, maybe I made a mistake. Wait, no, wait the side opposite angle $B$ is $b$? Wait, no! Wait, in triangle notation, side $a$ is opposite angle $A$, side $b$ opposite angle $B$, side $c$ opposite angle $C$. Wait, the diagram: angle $A$ is at vertex $A$, angle $B$ at $B$, angle $C$ at $C$. So side $a$ is $BC$, side $b$ is $AC$, side $c$ is $AB$. Wait, the diagram shows side $BC$ is 7? Wait, no, the diagram has side $BC$ labeled 7? Wait, the user's diagram: at vertex $B$, side $BC$ is 7? Wait, the given information: $A = 23^\circ$, $b = 9.5$ (which is side $AC$), and we need to find $c$ (side $AB$). Wait, maybe I misassigned the Law of Sines. Let's re-express:

In triangle $ABC$, angle $A = 23^\circ$, angle $B = 32^\circ$, angle $C = 125^\circ$, side $AC = b = 9.5$, side $BC = a = 7$ (wait, the diagram shows side $BC$ as 7? Wait, the user's diagram: at $B$, the side to $C$ is 7, so side $BC = a = 7$, side $AC = b = 9.5$, side $AB = c$ (what we need to find). Then using Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Wait, that's the mistake! I thought side $b$ was opposite angle $B$, but no: side $a$ is opposite angle $A$ (BC), side $b$ opposite angle $B$ (AC), side $c$ opposite angle $C$ (AB). So angle $A = 23^\circ$, opposite side $a = 7$; angle $B = 32^\circ$, opposite side $b = 9.5$? Wait, no, the given information: $A = 23^\circ$, $b = 9.5$, and the diagram has side $BC = 7$ (so $a = 7$), angle $C = 125^\circ$. So let's correct:

Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

So $a = 7$, angle $A = 23^\circ$; $b = 9.5$, angle $B = 32^\circ$; $c = ?$, angle $C = 125^\circ$

First, verify if the Law of Sines holds for $a$ and $b$:

$\frac{7}{\sin 23^\circ} \approx \frac{7}{0.3907} \approx 17.91$

$\frac{9.5}{\sin 32^\circ} \approx \frac{9.5}{0.5299} \approx 17.93$ (close, considering rounding errors)

Then to find $c$: $c = \frac{\sin C \times b}{\sin B}$? Wait, no: $c = \frac{\sin C \times a}{\sin A}$? Wait, no, let's use the correct ratio. Since $\frac{c}{\sin C} = \frac{b}{\sin B}$, so $c = \frac{b \times \sin C}{\sin B}$

We have $b = 9.5$, $\sin C = \sin 125^\circ \approx 0.8192$, $\sin B = \sin 32^\circ \approx 0.5299$

So $c = \frac{9.5 \times 0.8192}{0.5299} \approx \frac{7.7824}{0.5299} \approx 14.7$ (but wait, if $a = 7$, angle $A = 23^\circ$, then $\frac{7}{\sin 23^\circ} \approx 17.91$, so $c = 17.91 \times \sin 125^\circ \approx 17.91 \times 0.8192 \approx 14.7$, which matches. So that's correct.

Answer:

$\approx 14.7$ (depending on rounding, maybe to one decimal place, or as needed. If we calculate more precisely:

$\sin 23^\circ \approx 0.3907311285$

$\sin 32^\circ \approx 0.5299192642$

$\sin 125^\circ = \sin(55^\circ) \approx 0.8191520443$

$\frac{7}{\sin 23^\circ} = \frac{7}{0.3907311285} \approx 17.915$

Then $c = 17.915 \times \sin 125^\circ \approx 17.915 \times 0.8191520443 \approx 14.7$)

So the length of side $c$ is approximately $\boxed{14.7}$ (or more precise, but likely rounded to one decimal place).