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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. $14$

We  use following two properties of triangles

1)for otbuse triangle pythagoras thm says that

c^2>a^2+b^2

2)sum of two sides  greater than third side

case 1:so if x is bigger

x<15+8 ie 23

x can take values 15 16 17 18 19 20 21 22

15^2+8^2=17^2

so x can take values 18 19 20 21 22 only

case 2:if x is not bigger

15<x+8    x>7   ie  8 9 10 11 12 13 14

15^2>x^2+8^2

x^2<225-64

x^2<161

so x take values  8 9 10 11 12

so total 10 values possible for x
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