Chapter 12.2, Problem 55E

To determine

Prove that a right triangle whose sides are in arithmetic progression is similar to a $3-4-5$ triangle.

Explanation of Solution

Given:

It is given in the question that the sides be $3-4-5$ .

Concept Used:

In this, use the concept taking the sides of the triangle be $(a-d),a,(a+d)$ and make them in arithmetic progression.

Proof: Let the side of right angle triangle be : $(a-d),a,(a+d)$

Here all the side of the right angle triangle are in arithmetic progression with the common difference equal to ‘ d’ .

Since, the given triangle is right angle so, it will follow Pythagoras theorem ,

  $\begin{array}{l}$ {(a-d)}^2+a^2={(a+d)}^2$a^2-2ad+d^2+a^2=a^2+2ad+d^2\\ a^2=4ad\\ a=4d\end{array}$

Now, sides of right angle triangle will be : $(a-d)=3d,a=4d,(a+d)=5d$

So, the sides of triangle are in Arithmetic Progression - $3:4:5$