Sequence And Series Question 309
Question: If the altitudes of a triangle are in A.P., then the sides of the triangle are in
[EAMCET 2002]
Options:
A) A.P.
B) H.P.
C) G.P.
D) Arithmetico-geometric progression
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ P_1,P_2,P_3 $ be altitudes from P, Q and R $ P_1=c\sin Q=\lambda bc $ , $ P_2=a,\sin R=\lambda ca $ $ P_3=b\sin P=\lambda ab $ $ [ \therefore \frac{\sin P}{a}=\frac{\sin Q}{b}=\frac{\sin R}{c}=\lambda ] $
Þ $ P_1,P_2,P_3 $ are in A.P.
Þ $ \lambda bc,\lambda ca,\lambda ab $ are in A.P.
Þ $ bc,ca,ab $ are in A.P.
Þ $ \frac{abc}{a},\frac{abc}{b},\frac{abc}{c} $ are in A.P $ \frac{1}{a},,\frac{1}{b},,\frac{1}{c} $ are in A.P.
$ \therefore a,b,c $ are in H.P. i.e., sides of the triangle are in H.P.
BETA
