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

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q 2: why is 3,4,4 not possible??????

q1: why is 5,5,5 not among the possible options??