When a pendulum swings 40 degrees from the vertical, the bob moves 20 cm horizontally and 7.3 cm vertically. What is the length of the pendulum?

When a pendulum swings 40 degrees from the vertical, the
bob moves 20 cm horizontally and 7.3 cm vertically. What is the length of the
pendulum?

Lets name the point where the pendulum hangs
from A, the point where the bob rests directly beneath A we will call B, and the point
after displacement of the bob C.

Drop a perpendicular from
C with length 7.3cm and call the terminal point D. Connect D to B with a segment -- then
triangle BCD is a right triangle. Using the pythagorean theorem, we find the hypotenuse
BC to be approximately 21.29cm.

But BC is the base of an
isosceles triangle whose legs are the length of the pendulum. Drop an altitude from A to
BC and name the point of intersection M. The altitude of an isosceles triangle is also a
median, and an angle bisector.

Consider the triangle ABM.
It is a right triangle,with one leg opposite angle A having length 10.65cm (1/2 of
21.29cm). The angle MAB has measure 20 degrees. Thus `sin20=10.65/(BA)` , from the right
triangle definition of sine. Then `BA ~~ 31.14` cm, where
BA is the length of the pendulum.