Sum to infinity of geometric s

In the figure, triangle ABC is a right angled triangle with angle A = 30° and AC = 30 cm. d1, d2, d3, ... are the lengths of perpendiculars to AB or AC.
a) Find the values of d1 and d2.
b) Find the value of d1+d2+d3+...if the perpendiculars are drawn infinitely.


a)
In ΔADC:
sin30° = d1/AC
(1/2) = d1/(30 cm)
d1 = 15 cm

In ΔAED:
cos30° = AD/AC
(√3)/2 = AD/30 cm
AD = 15√3 cm

The perpendicular from D to AC cuts AC at E.
In ΔADE:
sin30° = d2/AD
(1/2) = d2/(15√3 cm)
d2 = 15(√3)/2 cm


b)
d1, d2, d3 …… is a G.P. with
first term, a = 15
and common ratio, r = (√3)/2 < 1

d1 + d2 + d3 + …… + to infinite term
= {15 + 15[(√3)/2] + 15[(√3)/2]² + …… + to infinite term} cm
= 15/[1 - (√3)/2] cm
= 30/[2 - (√3)] cm
= 30[2 + (√3)]/[2 - (√3)][2 + (√3)] cm
= [60 + 30√3]/[(2)² - (√3)²] cm
= (60 + 30√3) cm