# Question Video: Using a Triangle of Forces to Solve Equilibrium Problems Mathematics

Three coplanar forces 𝐹₁, 𝐹₂, and 𝐹₃ are acting on a body in equilibrium. Their triangle of forces forms a right triangle. Given that 𝐹₁ = 5 newtons and 𝐹₂ = 13 newtons, find the magnitude of 𝐹₃.

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### Video Transcript

Three coplanar forces 𝐹 sub one, 𝐹 sub two, and 𝐹 sub three are acting on a body in equilibrium. Their triangle of forces forms a right triangle as shown. Given that 𝐹 sub one is equal to five newtons and 𝐹 sub two is equal to 13 newtons, find the magnitude of 𝐹 sub three.

Remember, when three coplanar forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the adjacent sides of a triangle taken in order. We actually have the triangle of forces drawn for us, and we know the magnitude of two of the forces. 𝐹 sub one is equal to five newtons and 𝐹 sub two is equal to 13 newtons. This triangle now represents the relative magnitude of each of our forces. And since it forms a right triangle, we can find the magnitude of the third force by using the Pythagorean theorem. This tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. If we let the hypotenuse be equal to 𝑐, then we say that 𝑎 squared plus 𝑏 squared equals 𝑐 squared.

In this case, the longest side in our triangle is the side represented by the 13-newton force. And so, using the magnitudes we’ve been given and letting the magnitude of 𝐹 sub three be equal to 𝑏 newtons, we can say that five squared plus 𝑏 squared equals 13 squared. That is, 25 plus 𝑏 squared equals 169. And subtracting 25 from both sides of this equation, we find 𝑏 squared is equal to 144. To solve for 𝑏, we simply need to find the square root of both sides of this equation. Now, usually we would find both the positive and negative square root of 144. But since this represents a magnitude, we know it absolutely must be positive. And so 𝑏 is equal to the square root of 144, which is 12. Given that 𝐹 sub one is five newtons and 𝐹 sub two is 13 newtons then, we can say the magnitude of 𝐹 sub three is 12 newtons.