# 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?

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## 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.