KVPY SX SB Exam 2015 Question Paper

Question : 20 of 120

Marks: +1, -0

The sides of a right-angled triangle are integers. The length of one of the sides is 12. The largest possible radius of the incircle of such a triangle is -

Validate

Solution:

Clearly

x ^2 - y ^2 = 144

(x - y) (x + y) = 144

x, y ∊ I

So factorsize 144 into two even factors

Case I	Case I	Case I
x + y = 72	x + y = 36	x + y = 18
x – y = 2	x – y = 9	x – y = 8
x = 37	x = 20	x = 13
y = 35	y = 16	y = 5

$r = {∆} /s$	$r = {∆} /s$	$r = {∆} /s$
$r ={1\/ 2 ×12 ×35}/({12 +35 +37}/2)$	$r ={1\/ 2 ×12 ×16}/({16 +20+12}/2)$	r = 2
$r={12×35}/84$	$r={12×16}/48$
r = 5	r = 4

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