Question

solve for c.
a = 12
law of sines: \\(\frac{\sin a}{a} = \frac{\sin b}{b} = \frac{\sin c}{c}\\)
round your answer to the nearest hundredth.
c = ?

Explanation:

Step1: Find angle B

In a triangle, the sum of angles is \(180^\circ\). So \( \angle B = 180^\circ - 73^\circ - 32^\circ = 75^\circ \).

Step2: Apply Law of Sines

Law of Sines: \( \frac{\sin A}{a} = \frac{\sin C}{c} \)? Wait, no, correct: \( \frac{\sin A}{a} = \frac{\sin C}{c} \)? Wait, angle A is \(32^\circ\), side a is 12 (opposite angle A? Wait, no, in the triangle, side a is opposite angle A? Wait, label: side a is BC? Wait, no, standard notation: side a is opposite angle A, side b opposite angle B, side c opposite angle C. Wait, in the diagram, side a is AB? Wait, no, the diagram: angle A is \(32^\circ\), angle C is \(73^\circ\), side a (opposite angle A) is BC? Wait, no, the label: a = 12 is side BC? Wait, no, the diagram: vertex B, C, A. Side a is opposite angle A, so side a is BC. Wait, angle A is \(32^\circ\), angle C is \(73^\circ\), side a (BC) is 12? Wait, no, the problem says a = 12, and angle at C is \(73^\circ\), angle at A is \(32^\circ\). Wait, let's re-express: angle A = \(32^\circ\), angle C = \(73^\circ\), side a (opposite angle A) is BC? Wait, no, standard notation: side a is opposite angle A, side b opposite angle B, side c opposite angle C. So angle A: \(32^\circ\), opposite side a? Wait, no, in the diagram, side labeled a = 12 is between B and C? Wait, no, the diagram: B---c---A, C is the vertex with \(73^\circ\), B is the other vertex. So side a is AB? Wait, maybe I mixed up. Let's use the Law of Sines correctly. The Law of Sines is \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \), where a is opposite angle A, b opposite angle B, c opposite angle C. So in the diagram: angle A = \(32^\circ\), angle C = \(73^\circ\), so angle B = \(75^\circ\) (as calculated). Side a (opposite angle A) is BC, side c (opposite angle C) is AB? Wait, no, the side we need to find is c (BC? Wait, the question is to solve for c, which is the side between B and A, labeled c. Wait, the diagram: side c is BA? No, the side labeled c is BC? Wait, the diagram: B---c---A, so side c is BC? Wait, no, the label: c is the side from B to A? Wait, maybe the notation is: side a is AC (length 12), angle at C is \(73^\circ\), angle at A is \(32^\circ\). Then angle B is \(75^\circ\). Then Law of Sines: \( \frac{\sin A}{a} = \frac{\sin C}{c} \)? Wait, angle A is \(32^\circ\), side a is BC? No, side a is AC (length 12), opposite angle A? Wait, angle A is \(32^\circ\), opposite side a (BC). Wait, this is confusing. Wait, the problem says "Solve for c", and the Law of Sines is given as \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \). Wait, maybe angle A is \(32^\circ\), side a is 12 (opposite angle A), angle C is \(73^\circ\), and we need to find side c (opposite angle C). Wait, no, that would be \( \frac{\sin A}{a} = \frac{\sin C}{c} \), so \( c = \frac{a \cdot \sin C}{\sin A} \). Wait, angle A is \(32^\circ\), side a is 12 (opposite angle A), angle C is \(73^\circ\), so \( c = \frac{12 \cdot \sin 73^\circ}{\sin 32^\circ} \). Let's calculate that. \( \sin 73^\circ \approx 0.9563 \), \( \sin 32^\circ \approx 0.5299 \). So \( c = \frac{12 \times 0.9563}{0.5299} \approx \frac{11.4756}{0.5299} \approx 21.66 \). Wait, but let's check angle B: \(180 - 32 - 73 = 75^\circ\). Alternatively, maybe side a is opposite angle A (32°), side c is opposite angle C (73°), so Law of Sines: \( \frac{\sin A}{a} = \frac{\sin C}{c} \), so \( c = \frac{a \sin C}{\sin A} \). Plugging in: \( a = 12 \), \( \sin A = \sin 32^\circ \approx 0.5299 \), \( \sin C = \sin 73^\circ \approx 0.9563 \). So \( c = \frac{12 \times 0.9563}{0.5299} \approx 21.66 \). Wait, but let's confirm the angle labels. In the diagram, angle at A is 32°, angle at C is 73°, so side a (opposite angle A) is BC, side c (opposite angle C) is AB? No, maybe the side labeled a is AC (length 12), opposite angle A (32°), so AC = 12, angle A = 32°, angle C = 73°, then side c (AB) is what we need? Wait, no, the problem says "Solve for c", and the side labeled c is BC? Wait, the diagram has B---c---A, so c is the length of BA? No, the label c is between B and A, so c is BA. Wait, maybe I messed up the notation. Let's start over.

Given triangle ABC, with angle A = 32°, angle C = 73°, side a (opposite angle A) = 12? Wait, no, the problem says "a = 12" and in the diagram, a is the side from B to C? Wait, the diagram: vertex B, C, A. Side a is BC, length 12? No, the label "a = 12" is next to side BC? Wait, no, the diagram: angle at C is 73°, angle at A is 32°, side a = 12 is between B and C? Wait, maybe the correct notation is: angle A = 32°, angle C = 73°, side a (opposite angle A) is BC, length 12? No, that can't be. Wait, the Law of Sines is \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \). Let's use angle A = 32°, angle C = 73°, so angle B = 75°. Then, side a is opposite angle A (32°), side c is opposite angle C (73°). So if side a is 12, then \( \frac{\sin 32°}{12} = \frac{\sin 73°}{c} \), so \( c = \frac{12 \times \sin 73°}{\sin 32°} \). Calculating:

\( \sin 32° \approx 0.5299 \)

\( \sin 73° \approx 0.9563 \)

\( c = \frac{12 \times 0.9563}{0.5299} \approx \frac{11.4756}{0.5299} \approx 21.66 \)

So the length of c is approximately 21.66.

Answer:

\( \boxed{21.66} \) (rounded to nearest hundredth)