Question

Condition for an acute-angled triangle with sides a, b, and c is 
a2+b2>c2 (where c is the longest side).
If y is an integer, then how many acute-angled triangles with sides 7 cm, 12 cm, and y cm exist? ​

Answer 1

Given :  Condition for an acute-angled triangle with sides a, b, and c is  

a²+b²>c²(where c is the longest side).

To Find : y is an integer, then how many acute-angled triangles with sides 7 cm, 12 cm, and y cm exist

Solution:

a² + b²  > c²

a = 7

b = 12

Case 1 :

c = y  is longest side => y ≥ 12

Hence

7² + 12² > y²

=> 49 + 144 > y²

=> 193 > y²

y = 12  , 13  

Case 2 :   12 is the longest side

y < 12

=> 7² +  y² >  12²

=> y² > 95

=> y = 10 ,11

Possible values of y are  

10 , 11 , 12 & 13

4 acute triangles are possible

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