### Video Transcript

π΄π΅πΆ is a triangle, where the
side π΄π΅ equals 20 centimeters, π΅πΆ equals 25 centimeters, and πΆπ΄ equals 15
centimeters. And forces of magnitudes 120, 150,
and 90 newtons are acting along π΄π΅, π΅πΆ, and πΆπ΄, respectively, where the system
is equivalent to a couple. Determine the magnitudes of the two
parallel forces that would make the system in equilibrium when acting at π΅ and πΆ
perpendicularly to π΅πΆ.

Letβs start with a diagram of the
scenario. We have the triangle π΄π΅πΆ with
side lengths π΄π΅ equals 20 centimeters, π΅πΆ equals 25 centimeters, and πΆπ΄ equals
15 centimeters. The force along π΄π΅ is 120
newtons. The force along π΅πΆ is 150
newtons. And the force along πΆπ΄ is 90
newtons. We have two parallel forces πΉ one
and πΉ two, acting from the points π΅ and πΆ and perpendicular to π΅πΆ. The other forces acting on the
triangle are equivalent to a couple. So in order to bring the system to
equilibrium, the two forces acting at π΅ and πΆ must also be equivalent to a couple
of equal magnitude and opposite direction. So πΉ one is equal to πΉ two. Letβs call this magnitude just
πΉ.

To find the magnitude πΉ, we need
to find the magnitude of the couple formed by the other forces acting on the
triangle. To do this, we can calculate the
moments of all three forces about a point of our choice and then add them
together. Letβs calculate the moments of the
three forces about the point π΄, taking the convention that counterclockwise moments
are positive.

Recall that the magnitude of the
moment π of a force πΉ acting from a point π is equal to the magnitude of the
force πΉ multiplied by the perpendicular distance π between the pivot point π and
the line of action of πΉ. The lines of action of the forces
of 90 newtons and 120 newtons pass through the point π΄. So their perpendicular distances
from π΄ are both zero. And hence, their moments about π΄
are also both zero. The only force that has a nonzero
moment about π΄ is the force of 150 newtons. We need to find the perpendicular
distance from the line of action of this force to the point π΄. This is equivalent to finding the
altitude of the triangle with the line π΅πΆ as the base.

Recall that the altitude of a
scalene triangle, that is, a triangle with three sides of different lengths, is
given by two over π multiplied by the square root of π times π minus π times π
minus π times π minus π, where π is the semiperimeter of the triangle. π is the base of the triangle. And π and π are the other two
sides of the triangle. The semiperimeter, π , is half the
perimeter of the triangle, which is equal to 15 plus 20 plus 25 over two, which is
equal to 30 centimeters. π is the length of the base line
π΅πΆ, which is 25 centimeters. π and π are the lengths of the
other two sides, which are 20 centimeters and 15 centimeters.

So, the altitude of the triangle,
which is equivalent to the perpendicular distance π between the pivot point π΄ and
the line of action of the force of 150 newtons, is given by two over 25 multiplied
by the square root of 30 times 30 minus 20 times 30 minus 25 times 30 minus 15. This comes to the nice round
number, 12 centimeters. The magnitude of the moment π is
therefore given by the magnitude of the force, 150, multiplied by this perpendicular
distance, 12, which is equal to 1800 newton-centimeters. Therefore, the magnitude of the
resultant moment on the system is 1800 newton-centimeters. And the magnitude of the couple of
the two parallel forces introduced at the points π΄ and π΅ must therefore also be
1800 newton-centimeters.

Recall that the magnitude of a
couple acting at two points π΅ and πΆ with the forces at an angle π between their
lines of action and the line π΅πΆ is equal to the magnitude of one of the forces,
πΉ, multiplied by the length of the line π΅πΆ multiplied by its sin π. In this case, the two forces are
perpendicular to the line π΅πΆ. So the angle π is equal to 90
degrees, and therefore sin π is equal to one. We can therefore simplify this to
πΉ times the length of the line π΅πΆ. We can rearrange this equation for
πΉ to give πΉ equals the magnitude of the couple, π π΅πΆ, over the length of the
line π΅πΆ. This is equal to 1800 over 25. Therefore, the magnitudes of the
two parallel forces that would make the system in equilibrium are both 72
newtons.