Question

The sides of a triangle are distinct positive integers in an arithmetic progression. If the smallest side is $10$, the number of such triangles is$?$

• A. $8$
• B. $9$
• C. $10$
• D. Infinitely many

Answer 1

Answer: (B)

$9$

Given that sides of a triangle are in arithmetic progression.

Let $a,a+d,a+2d$ be the sides of triangle.

Given $a=10$, so the sides of a triangle are $10,10+d,10+2d$

The sum of any two sides of a triangle must be greater than the third side.

$\Rightarrow 10+10+d>10+2d$

$\Rightarrow d<10$

We know that $d>0$ because the sides of triangle are distinct

$\Rightarrow 0<d<10$

Therefore, $9$ such triangles are possible.

So, the correct option is $B$