Closed vector triangles

Hang a $30\ \text{N}$ lamp from a cable and hold it to one side with a horizontal $40\ \text{N}$ pull. The tension in the cable comes out at exactly $50\ \text{N}$.

That's no coincidence: the three forces make a 3-4-5 right-angled triangle.

Forces add like any vectors: draw them tip to tail, and the resultant runs from the very start to the very end.

In equilibrium the resultant is zero, so the end lands back on the start. Three forces in equilibrium make a closed triangle.

1. Sketch the free-body diagram so you know each force's direction.
2. Choose a scale, for example 1 cm to 10 N.
3. Draw a known force (usually the weight) as an arrow to scale.
4. From its tip, draw a line in the direction of the second force.
5. From the tail of the first arrow, draw a line in the direction of the third force. Where the two lines cross closes the triangle.
6. Measure the sides and angles, and convert back using your scale.

The arrows must chase each other round the triangle, all in the same sense.