Geometry-7 (CAT-2008)

Consider obtuse angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer, then how many such triangles exist?
(1)5      (2)21         (3)10           (4) 15          (5) 14
Solution follows here:

Solution:

Here two theorems come into picture.

One is Pythagoras for obtuse triangle: c²>a²+b², where c is the biggest side of the three

Second one is Triangle inequality theorem: Sum of two sides of a triangle must be greater than the measure of the third side.  c < a+b, where c is the biggest side of the three.

Case-I:  x is the biggest side:

All possibilities of x according to triangle inequality theorem, are 22,21,20,19,18,17,16,15.

But 8²+15² = 64+225 = 289 = 17², hence as per Pythagoras, we can consider values above 17 only. x=22,21,20,19,18

Case-II: 15 is the biggest side:

All possibilities of x according to triangle inequality theorem, are 8,9,10,11,12,13,14.

But 15² - 8² = 225-64 = 161 < 13², hence as per Pythagoras, we can consider values less than 13 only. x=8,9,10,11,12

All possible values of x  = 8,9,10,11,12,18,19,20,21,22

Total 10 values 

Answer(3)