Have given below the solutions to the questions on geometry. These can also be found at our blog at

http://iimcat.blogspot.com/2011/08/cat-geometry-solutions.html

1. Perimeter of a triangle with integer sides is equal to 15.
How many such triangles are possible?

This is just a counting question, with the caveat that sum of two
sides should be greater than the third. Let us assume a

__< __ b

__< __ c

a = 1, Possible triangle 1, 7, 7

a = 2, possible triangle 2, 6, 7

a = 3, possible triangles 3, 6, 6 and 3, 5, 7

a = 4, possible triangles 4, 4, 7 and 4, 5, 6

There are totally 6 triangles possible

2. Triangle ABC has integer sides x, y, z such that xz = 12. How
many such triangles are possible?

xz = 12

x,z can be 1, 12 or 2, 6 or 3, 4

Possible triangles

1-12-12

2-6-5; 2-6-6; 2-6-7

3-4-2; 3-4-3; 3-4-5; 3-4-6

8 triangles are possible

3. Triangle has sides a^2, b^2 and c^2. Then the triangle with
sides a, b, c has to be - a) Right angled b) Acute-angled c) Obtuse
angled d) can be any of these three

Assuming a

__< __ b

__< __ c, we have a^2 +
b^2 > c^2. This implies the triangle with sides a, b, c has to be
acute-angled