Question

In a triangle ABC , sinA , sinB , sinC are in AP then
The altitudes are in AP
The altitudes are in HP
The angles are in AP
The angles are in HP

Answer 1

Hint - In such questions representing all the three sides a , b , c in terms of a constant like area of the triangle would be used to find the relation between all the sides . Using what is given in the question with the sine formula would give you the desired result .

Complete step-by-step answer:

Let us consider the triangle ABC with sides a, b, c and altitude $h_1,h_2,h_3$ upon the base BC , AC , AB respectively .
Let the area of the triangle ABC be R
R = ${\displaystyle \frac{1}{2}}$$\left(h_1\times a\right)$ = ${\displaystyle \frac{1}{2}}$ $\left(h_2\times b\right)$ = ${\displaystyle \frac{1}{2}}$ $\left(h_3\times c\right)$
( area of the triangle = ${\displaystyle \frac{1}{2}}$ base $\times$ height )
$\Rightarrow$ $a={\displaystyle \frac{2R}{h_1}};b={\displaystyle \frac{2R}{h_2}};c={\displaystyle \frac{2R}{h_3}}$

Now using the sine formula
${\displaystyle \frac{a}{\sin A}}={\displaystyle \frac{b}{\sin B}}={\displaystyle \frac{c}{\sin C}}$ = k (constant)
$\Rightarrow$ sinA = ${\displaystyle \frac{a}{k}}$ ; sinB = ${\displaystyle \frac{b}{k}}$ ; sinC = ${\displaystyle \frac{c}{k}}$
     $$$$
Now according to the question sinA , sinB , sinC are in AP
$\Rightarrow$ ${\displaystyle \frac{a}{k}},{\displaystyle \frac{b}{k}},{\displaystyle \frac{c}{k}}$ are in AP

Now , putting values of a , b , c from above ,
  ${\displaystyle \frac{1}{h_1}}{\displaystyle \frac{2R}{k}};{\displaystyle \frac{1}{h_2}}{\displaystyle \frac{2R}{k}};{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{2R}{k}}$ are in AP
$\Rightarrow$ ${\displaystyle \frac{1}{h_1}};{\displaystyle \frac{1}{h_2}};{\displaystyle \frac{1}{h_3}}$ are in AP ( cancelling out the constants )
Therefore ,
$h_1,h_2,h_3$ are in HP
Altitudes are in HP .

Note -
In these questions it is suitable to find an indirect method rather than to directly solve what's given . Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle. Note that you should try and keep full accuracy until the end of your calculation to avoid errors .
$\Rightarrow$ ${\displaystyle \frac{1}{h_1}};{\displaystyle \frac{1}{h_2}};{\displaystyle \frac{1}{h_3}}$ are in AP ( cancelling out the constants )
Therefore ,
$h_1,h_2,h_3$ are in HP
Altitudes are in HP .