# Question Video: Finding the Ratio between the Forces in System of Three Forces Acting through a Triangle Mathematics

In the figure, three forces of magnitudes 𝐅₁, 𝐅₂, and 𝐅₃ newtons meet at a point. The lines of action of the forces are parallel to the sides of the right triangle. Given that the system is in equilibrium, find the ratio of 𝐅₁ : 𝐅₂ : 𝐅₃.

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### Video Transcript

In the figure, three forces of magnitudes 𝐅 sub one, 𝐅 sub two, and 𝐅 sub three newtons meet at a point. The lines of action of the forces are parallel to the sides of the right triangle. Given that the system is in equilibrium, find the ratio of 𝐅 sub one to 𝐅 sub two to 𝐅 sub three.

Remember, when the three forces are in equilibrium, the magnitudes of the forces are proportional to the side lengths of the triangle. Let’s begin by finding the length of the third side. We can label the vertices of the triangle as shown, meaning we want to find the length of the hypotenuse, 𝐵𝐶. Let’s use the Pythagorean theorem, so 𝐵𝐶 squared equals 87 squared plus 208.8 squared. That gives us 𝐵𝐶 squared equals 51166.44. Taking the positive square root gives us the length 𝐵𝐶 to be 226.2 centimeters.

Since the triangle is in equilibrium, we know that the ratios of the forces and the side lengths they are parallel to are all equal. We can therefore also say that the ratio of two of the forces must be equal to the ratio of the respective side lengths. Since 𝐴𝐵 is 87 centimeters and 𝐴𝐶 is 208.8 centimeters, we can divide these to get five twelfths. In a similar way, we can find the ratio of 𝐅 sub one and 𝐅 sub three by dividing 𝐴𝐵 by 𝐵𝐶. That gives us 𝐅 sub one over 𝐅 sub three equals five over 13. Since the numerator is the same for each fraction, we can create the required ratio. The ratio of 𝐅 sub one to 𝐅 sub two to 𝐅 sub three is five to 12 to 13.