### Video Transcript

In the figure, three forces of
magnitudes π
sub one, π
sub two, and π
sub three 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 the ratio of π
sub one to π
sub two to π
sub three.

Remember, when the three forces
are in equilibrium, the magnitudes of the forces are proportional to the side
lengths of the triangle. Letβs begin by finding the
length of the third side. We can label the vertices of
the triangle as shown, meaning we want to find the length of the hypotenuse,
π΅πΆ. Letβs use the Pythagorean
theorem, so π΅πΆ squared equals 87 squared plus 208.8 squared. That gives us π΅πΆ squared
equals 51166.44. Taking the positive square root
gives us the length π΅πΆ to be 226.2 centimeters.

Since the triangle is in
equilibrium, we know that the ratios of the forces and the side lengths they are
parallel to are all equal. We can therefore also say that
the ratio of two of the forces must be equal to the ratio of the respective side
lengths. Since π΄π΅ is 87 centimeters
and π΄πΆ is 208.8 centimeters, we can divide these to get five twelfths. In a similar way, we can find
the ratio of π
sub one and π
sub three by dividing π΄π΅ by π΅πΆ. That gives us π
sub one over
π
sub three equals five over 13. Since the numerator is the same
for each fraction, we can create the required ratio. The ratio of π
sub one to π
sub two to π
sub three is five to 12 to 13.