question_answer

If the altitudes of a triangle are in A.P., then the sides of the  triangle are in [EAMCET 2002]

A) A.P.

B) H.P.

C) G.P.

D) Arithmetico-geometric progression

Correct Answer: B

Solution :

  Let  \[{{P}_{1}},{{P}_{2}},{{P}_{3}}\] be altitudes from P, Q and R \[{{P}_{1}}=c\sin Q=\lambda bc\], \[{{P}_{2}}=a\,\sin R=\lambda ca\] \[{{P}_{3}}=b\sin P=\lambda ab\]   \[\left[ \therefore \frac{\sin P}{a}=\frac{\sin Q}{b}=\frac{\sin R}{c}=\lambda  \right]\] 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.