5. How many perpendicular lines must be drawn in a triangle to prove the Law of Sines? 0 1

To prove the **Law of Sines** in a triangle, you need to draw **one perpendicular line**. Here's why: ### **Understanding the Law of Sines** The Law of Sines states that for any triangle: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] where: - \( a, b, c \) are the lengths of the sides opposite angles \( A, B, C \) respectively. ### **Proof Outline Using One Perpendicular** 1. **Draw the Perpendicular (Altitude):** - Choose one vertex of the triangle and draw a perpendicular line (altitude) to the opposite side. This divides the triangle into two right-angled triangles. 2. **Apply Trigonometric Ratios:** - In each right-angled triangle, use the sine function to relate the sides and angles: \[ \sin A = \frac{\text{opposite side}}{\text{hypotenuse}} \] - This leads to expressions that relate the sides of the original triangle to the sines of their opposite angles. 3. **Derive the Law of Sines:** - By manipulating these expressions, you arrive at the proportionality that defines the Law of Sines. ### **Conclusion** Only **one perpendicular line** is necessary to establish the necessary right-angled triangles for applying trigonometric ratios, which then lead to the proof of the Law of Sines. **Answer:** 1