5. How many perpendicular lines must be drawn in a triangle to prove the Law of Sines? 0 2

To prove the Law of Sines, we need to understand the relationship between the angles and sides of a triangle. The Law of Sines states that: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] where \(a\), \(b\), and \(c\) are the lengths of the sides opposite to angles \(A\), \(B\), and \(C\) respectively. To derive this law, we can use the concept of perpendicular lines, specifically the height of the triangle from one vertex to the opposite side. ### Steps to Prove the Law of Sines: 1. **Draw a Triangle**: Consider triangle \(ABC\) with sides \(a\), \(b\), and \(c\) opposite to angles \(A\), \(B\), and \(C\) respectively. 2. **Draw a Perpendicular Line**: From vertex \(A\), draw a perpendicular line to side \(BC\), meeting it at point \(D\). This line represents the height \(h_a\) from vertex \(A\) to side \(BC\). 3. **Use Right Triangles**: In triangle \(ABD\) and triangle \(ACD\), we can use the definitions of sine: - In triangle \(ABD\): \[ \sin(B) = \frac{h_a}{b} \implies h_a = b \sin(B) \] - In triangle \(ACD\): \[ \sin(C) = \frac{h_a}{c} \implies h_a = c \sin(C) \] 4. **Set the Heights Equal**: Since both expressions equal \(h_a\), we can set them equal to each other: \[ b \sin(B) = c \sin(C) \] 5. **Rearranging**: This can be rearranged to show the relationship between the sides and angles: \[ \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] 6. **Repeat for Other Angles**: To complete the proof for all three sides, you would repeat the process for the other two angles, drawing perpendiculars from vertices \(B\) and \(C\) to the opposite sides. ### Conclusion: To prove the Law of Sines, you need to draw **2 perpendicular lines** from two vertices to the opposite sides. Therefore, the answer is: **2**