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Lesson Worksheet: Triangle of Forces Mathematics

In this worksheet, we will practice solving problems about the equilibrium of a particle under the action of three forces meeting at a point using the resultant of forces or triangle of force method.


Which of the following represents the triangle of forces rule?

  • A𝐹𝐴=𝐹𝐶=𝐹𝑂sinsinsin


Complete the following: If three coplanar and equilibrium forces are represented by the sides of a triangle taken in one cyclic order, then the lengths of the sides of the triangle are proportional to .

  • Athe squares of the magnitudes of the forces
  • Bthe magnitude of the forces
  • Cthe square of the angles
  • Dthe angles


Anthony is attempting a mechanics problem in which three coplanar forces F, F, and F are acting on a body. He needs to determine whether the body is in equilibrium or not. He remembers his teacher saying something about checking whether he could arrange the forces into a triangle. So, he draws the shown figure.

Anthony concludes that the three forces are in equilibrium. Is he correct?

  • AYes
  • BNo

Which of the following best describes what he has done?

  • AHe has put the forces in the wrong order. He should have started with the force represented by the longest arrow and worked his way to the shortest.
  • BHe has not paid attention to the direction of the forces. All the forces should meet head-to-tail. However, in his diagram, F and F meet head-to-head. Therefore, the forces do not actually form a triangle.
  • CHe has used the wrong method; a force triangle is not a valid way of checking for equilibrium.
  • DHe has done nothing wrong.


Three coplanar forces F, F, and F are acting on a body in equilibrium. Their triangle of forces forms a right triangle as shown.

Given that F=5newtons and F=13newtons, find the magnitude of F.


A body of weight 𝑊 is attached to a wall by a string of length 25 cm. It is held in equilibrium by the effect of a horizontal force of magnitude 93 g-wt that keeps the body 15 cm away from the wall. Determine 𝑇 and 𝑊.

  • A𝑇=155g-wt, 𝑊=124g-wt
  • B𝑇=55.8g-wt, 𝑊=116.25g-wt
  • C𝑇=55.8g-wt, 𝑊=69.75g-wt
  • D𝑇=155g-wt, 𝑊=69.75g-wt


In the figure, three forces of magnitudes 𝐹, 𝐹, and 𝐹 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 𝐹𝐹𝐹.

  • A51312
  • B51213
  • C12513
  • D13125


Use the triangle force rule to find the magnitude of force 𝐹 that is in equilibrium with two perpendicular forces of magnitudes 2 and 3 newtons.

  • A1 N
  • B1.5 N
  • C5 N
  • D13 N
  • E6 N


The body at point 𝐵 is at equilibrium. Find the magnitude of the force F in newtons, rounding your answer to two decimal places.


A body is under the effect of three forces of magnitudes 𝐹, 𝐹, and 36 newtons, acting in the directions 𝐴𝐵, 𝐵𝐶, and 𝐶𝐴, respectively, where 𝐴𝐵𝐶 is a triangle such that 𝐴𝐵=4cm, 𝐵𝐶=6cm, and 𝐴𝐶=6cm. Given that the system is in equilibrium, find 𝐹 and 𝐹.

  • A𝐹=54N, 𝐹=36N
  • B𝐹=36N, 𝐹=24N
  • C𝐹=24N, 𝐹=54N
  • D𝐹=24N, 𝐹=36N


A uniform rod of length 50 cm and weight 143 N is freely suspended at its ends from the ceiling by means of two perpendicular strings attached to the same point on the ceiling. Given that the length of one of the strings is 30 cm, determine the tension in each string.

  • A85.8 N, 114.4 N
  • B238.33 N, 114.4 N
  • C85.8 N, 178.75 N
  • D238.33 N, 178.75 N

This lesson includes 29 additional questions and 225 additional question variations for subscribers.

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