   Chapter 11.2, Problem 80E

Chapter
Section
Textbook Problem

A right triangle ABC is given with ∠ A = θ and | A C | = b . CD is drawn perpendicular to AB, DE is drawn perpendicular to BC, E F ⊥ A B , and this process is continued indefinitely, as shown in the figure. Find the total length of all the perpendiculars | C D | + | D E | + | E F | + | F G | + ... in terms of b and θ . To determine

To find:

The total length of all the perpendiculars CD+DE+EF+FG+ in terms of b and θ

Explanation

1) Concept:

Geometric series:

n=1arn-1=a+ar+ar2+

is convergent if r<1 and its sum is

n=1arn-1=a1-rr<1

2) Given:

A right triangle  ABC, A=  and AC=b, CD  AB, DE BC,  EF  AB

3) Calculation:

On right triangle CDA,

sinθ=CDb

CD=bsin θ  …………………. (i)

On right triangle DEC,

DE=CDsin θ

=bsin θsin θ……………since  CD=b sin θ

So,

DE=bsin2 θ  ……………….. (ii)

On right triangle EFD,

EF=DEsin θ

EF=bsin3 θ  ………….since  DE=bsin2 θ ………………. (iii)

A right triangle FGE,

FG=  EFsin θ

FG= bsin4 θ……………

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