**A game board is in the shape of a right triangle. Ther hypotenuse is 2in. Longer than the longer leg ,and the longer leg is 1in. Less than twice as long as the shorter leg. How long is each side of the game board?**

**Answer:**

Consider l, a and b are the sides of the triangle in which l is the hypotenuse, a is the longer side and b is the shorter side.

As per the given conditions, they can be expressed as

l=2+a;a=2*b-1;
From Pythogerus theorem of right angled triangle, the magnitudes of three sides can be expressed as
l^{2} =a^{2}+b^{2};
Substituting the values of “l” and “b” in terms of “a” gives,
(2+a)^{2} =a^{2} +((a+1)/2)^{2}};
4+a^{2}+4*a = a^{2} +(a^{2}+ 2

*a +1)/4;*

4+4a = (a^{2}+ 2

4+4

*a +1)/4 ;*

a^{2}/4-7a/2 -15/4 =0 ;

a^{2}/4-7

Further simplification of the equation gives,

a^{2} -14

*a -15 =0 ;*

For finding the roots for this equation, it can be expressed as

a^{2} -15a+1

For finding the roots for this equation, it can be expressed as

a^{2} -15

*a -15 =0;*

a(a-15)+1*(a-15) =0;

a

(a+1)

*(a-15) =0;*

Thus the solutions for a are

a=-1 and a=15

As “a” is the magnitude of the side of a triangle and thus cannot be negative, so

a=15

Substituting the value of a in the relations between a and l and a and b gives,

l= 2+a

= 17

a=2b-1

Thus the solutions for a are

a=-1 and a=15

As “a” is the magnitude of the side of a triangle and thus cannot be negative, so

a=15

Substituting the value of a in the relations between a and l and a and b gives,

l= 2+a

= 17

a=2

15= 2*b-1

b=8;

Thus sides of the triangle are 17, 15 and 8 units