### Video Transcript

A body is under the effect of three
forces of magnitudes πΉ sub one, πΉ sub two, and 36 newtons, acting in the
directions of line segments π΄π΅, π΅πΆ, and π΄πΆ, respectively, where triangle
π΄π΅πΆ is a triangle such that π΄π΅ equals four centimeters, π΅πΆ equals six
centimeters, and π΄πΆ equals six centimeters. Given that the system is in
equilibrium, find πΉ sub one and πΉ sub two.

We know that 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. So weβre going to represent the
three forces given using a triangle. But weβre told that they act in the
directions of the various sides of triangle π΄π΅πΆ. So weβll sketch triangle π΄π΅πΆ
first. Triangle π΄π΅πΆ looks a little
something like this, and we notice that the sides π΄πΆ and π΅πΆ are both six
centimeters in length. So itβs actually an isosceles
triangle.

Weβll now use this triangle to
sketch a triangle of forces representing πΉ sub one, πΉ sub two, and the 36-newton
force. The force with magnitude πΉ sub one
newtons acts in the direction of line segment π΄π΅. Then the force with magnitude πΉ
sub two acts in the direction of line segment π΅πΆ. Notice that the force with
magnitude πΉ sub two begins at the terminal point of our previous force. And so we have to begin our third
force at the terminal point of πΉ sub two. But we were told that this
36-newton force acts in the direction of the line segment π΄πΆ, not the line segment
πΆπ΄. However, since we know that the
magnitude of this force is 36 newtons, we can label it as shown. If we were considering the
direction of the force, we would need to consider that this would be the negative
direction of our original force. But for magnitudes which just
represent size, this is absolutely fine.

Weβre now ready to compare our
triangles. Since each of our forces acts in
the same direction as each side in our triangle π΄π΅πΆ, the two triangles must in
fact be similar. And so we can say that the
magnitudes of each of our forces must be directly proportional to the lengths of the
sides in triangle π΄π΅πΆ. So we can find force πΉ sub two
really easily. We know that the sides π΄πΆ and
π΅πΆ are equal in length. So this force and this force must
be equal in magnitude. And so πΉ sub two must be equal to
36 newtons. And then we have two different ways
that we can calculate the magnitude πΉ sub one.

One way is to say that the ratio of
the line segment π΄π΅ to the ratio of line segment π΄πΆ will be equal to the ratio
of the magnitude πΉ sub one to the magnitude 36 newtons. In other words, four divided by six
will give us the same outcome as πΉ sub one divided by 36. And whilst we could simplify the
fraction four-sixths, it doesnβt make a lot of sense to do this because weβre going
to multiply both sides of this equation by 36. Then we spot that 36 and six have a
common factor of six. So πΉ sub one will be equal to six
times four over one, which is simply equal to 24. And so πΉ sub one is 24
newtons. Itβs worth noting at this point
that we couldβve used scale factor to calculate the value of πΉ sub one.

Since the two triangles are
similar, we can deduce that one is an enlargement or a dilation of the other. And thus, the scale factor for
enlargement would be 36, thatβs one of the dimensions on our force triangle, divided
by six, the corresponding dimension on triangle π΄π΅πΆ. 36 divided by six is six. And so we can transform any
measurement on our triangle π΄π΅πΆ onto the dimensions of our force triangle by
multiplying by six. This means πΉ sub one would be
equal to four times six, which is once again 24. πΉ sub one is 24 newtons and πΉ sub
two is 36 newtons.