Question

Let S be the set of all right angled triangles with integer sides forming consecutive terms of an arithmetic progression. The number of triangles in S with perimeter less than 30 is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

2

Answer 2

Let the side of right angled triangles are
$x, x + 1, x + 2$
$\therefore (x+2)^2 = (x+1 )^2 +(x)^2$
$⇒ x^2 + 4x + 4 = x^2 +2x +1 + x^2$
$⇒ x ^2 − 2x− 3 = 0$
$⇒ (x-3 )(x+ 1 )= 0$
$⇒ x = 3, x ≠ − 1$
Perimeter of triangle $= 3 x + 3$
$3x + 3 < 30$
$x + 1 < 10$
$x < 9$
The possible triangle whose sides are
$(3, 4, 5), (6, 8, 10)$
$∴$ Two triangles are possible
Therefore, the Correct Option is (C) 2