Question

If the altitudes of a triangle are in A.P., then the sides of the triangle are in

• A. A.P
• B. H.P
• C. G.P
• D. Arithmetico-geometric progression

Answer 1

The correct option is B

H.P

Explanation for Correct Option:

Step1:Apply triangle formula

Let $a,b,c$ be altitudes and altitude corresponding to them are $x,y,z$ of a triangle.

Area of triangle$=\frac{1}{2}\times$base$\times$altitude

Area of triangle$=\frac{ax}{2}=\frac{by}{2}=\frac{cz}{2}$

$\Rightarrow ax=by=cz=p$

$\Rightarrow a=\frac{p}{x},b=\frac{p}{y},c=\frac{p}{z}$

Since the altitude $a,b,c$ is given in A.P.

$\Rightarrow 2b=a+c$

Step 2: Put the value of $2b$ in the below equation

$$\begin{gathered} \Rightarrow \frac{2p}{y}=\frac{p}{x}+\frac{p}{z} \\ \Rightarrow \frac{2}{x}=\frac{1}{y}+\frac{1}{z} \end{gathered}$$

$\therefore x,y,andz$ are in H.P.

Hence option (2) is the answer.