### Video Transcript

The diagram shows a force of magnitude πΉ and a force of magnitude 12 newtons which are parallel and the resulting force of magnitude π
. Given that π΄π΅ is equal to 72 centimeters and π΄πΆ is equal to 48 centimeters, determine the magnitudes of πΉ and π
.

Our first step is to mark on the information given about the lengths of π΄π΅ and π΄πΆ. We are told that π΄π΅ is equal to 72 centimeters and π΄πΆ is equal to 48 centimeters. This means that πΆπ΅ is equal to 24 centimeters as 72 minus 48 is 24. Now that weβve marked on the distances, letβs consider what we need to work out. We are asked to find the magnitudes of the forces πΉ and π
. One way to do this is to take moments about any point on our line.

We recall that the moment of a force is equal to the magnitude of the force multiplied by the perpendicular distance π from the point at which the force is acting to the point at which we are taking moments. Since all three forces are parallel and are acting perpendicular to the line segment π΄π΅, we can find these perpendicular distances directly from our diagram. We will begin by taking moments about point πΆ. We do this as the resulting force π
is acting at this point, and it will therefore have a moment equal to zero.

Convention dictates that moments acting in a counterclockwise direction are positive and moments acting in a clockwise direction are negative. And since the three forces result in the system being in equilibrium, we know that the sum of the moments is equal to zero. The 12-newton force is acting in a clockwise direction about point πΆ and will therefore have a negative moment. The moment of this force is equal to negative 12 multiplied by 48 as the magnitude of the force is 12 newtons and it is a perpendicular distance of 48 centimeters from πΆ.

The force πΉ is acting in a counterclockwise direction about point πΆ and will therefore have a positive moment. This is equal to πΉ multiplied by 24. Setting the sum of the moments equal to zero, we have negative 12 multiplied by 48 plus πΉ multiplied by 24 is equal to zero. This simplifies to negative 576 plus 24πΉ is equal to zero. We can divide through by 24 such that negative 24 plus πΉ equals zero. Adding 24 to both sides of this equation, we have πΉ is equal to 24. The magnitude of the force πΉ is therefore equal to 24 newtons.

We are now in a position where we can calculate the magnitude of the resulting force π
. Since the other two forces are parallel and act in the same direction, we know that the resulting force will be equal to their sum. π
is equal to πΉ plus 12. Substituting πΉ is equal to 24, we have π
is equal to 24 plus 12, which is equal to 36. The resulting force has magnitude 36 newtons.

We now have the two answers as required. πΉ is equal to 24 newtons and π
is equal to 36 newtons.