Have given below the solutions to the questions on basic geometry
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
Again, from comments,
a = 5, possible triangle is 5,5,5,
There are totally 7 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.
As pointed out in the comments section, I have missed the triangle 3-4-4.
There are totally 9 triangles.
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.
P.S: Big thanks to 'maniac' for pointing out the errors