A question based on triangles and sequence and series.

Question

The sides of a right angle triangle are in arithmetic progression if the triangle has area $24$. What is the length of the smallest side? I try to solve this problem by taking $c^2=a^2+b^2$ and $2b=a+c$ but was unable to proceed. This question had come in my country's JEE advanced examination for the year 2017.

Answer 1

Take the sides of the triangle to be $x+y$, $x$, $x-y$ (where $x$ and $y$ are positive numbers). Apply Pythagoras theorem, $(x+y)^2 = x^2+(x-y)^2$

$\Longrightarrow(x+y)^2-(x-y)^2=x^2$

$\Longrightarrow 4xy = x^2$

$\Longrightarrow x=4y$

$\therefore$ sides are in the ratio 3:4:5, let them be 3k,4k and 5k and use the area.

Hope it helps:)