### Video Transcript

Three coplanar forces ๐น sub one,
๐น sub two, and ๐น sub three are acting on a body in equilibrium. Their triangle of forces forms a
right triangle as shown. Given that ๐น sub one is equal to
five newtons and ๐น sub two is equal to 13 newtons, find the magnitude of ๐น sub
three.

Remember, 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. We actually have the triangle of
forces drawn for us, and we know the magnitude of two of the forces. ๐น sub one is equal to five newtons
and ๐น sub two is equal to 13 newtons. This triangle now represents the
relative magnitude of each of our forces. And since it forms a right
triangle, we can find the magnitude of the third force by using the Pythagorean
theorem. This tells us that, in a right
triangle, the sum of the squares of the two shorter sides is equal to the square of
the hypotenuse. If we let the hypotenuse be equal
to ๐, then we say that ๐ squared plus ๐ squared equals ๐ squared.

In this case, the longest side in
our triangle is the side represented by the 13-newton force. And so, using the magnitudes weโve
been given and letting the magnitude of ๐น sub three be equal to ๐ newtons, we can
say that five squared plus ๐ squared equals 13 squared. That is, 25 plus ๐ squared equals
169. And subtracting 25 from both sides
of this equation, we find ๐ squared is equal to 144. To solve for ๐, we simply need to
find the square root of both sides of this equation. Now, usually we would find both the
positive and negative square root of 144. But since this represents a
magnitude, we know it absolutely must be positive. And so ๐ is equal to the square
root of 144, which is 12. Given that ๐น sub one is five
newtons and ๐น sub two is 13 newtons then, we can say the magnitude of ๐น sub three
is 12 newtons.