Question 255064
x = 17.


17 - 14 = 3
17 - 13 = 4
17 - 12 = 5


345 triangle is a right triangle.


solve by use of pythatogrean formula.


a^2 + b^2 = c^2


x is the side of the equilateral triangle.


(x-12) = largest side which would be equal to c.


(x-13) = midsize side which would be equal to b.


(x-14) = smallest side which would be equal to a.


a^2 + b^2 = c^2 becomes:


(x-14)^2 + (x-13)^2 = (x-12)^2


this becomes


x^2 - 28x + 196 + x^2 - 26x + 169 = x^2 - 24x + 144


combine like terms to get:


2x^2 - 54x + 365 = x^2 - 24x + 144


subtract x^2 from both sides of equation to get:


x^2 - 54x + 365 = -24x + 144


add 24x to both sides of equation to get:


x^2 - 30x + 365  = 144


subtract 144 from both sides of equation to get:


x^2 - 30x + 221 = 0


factor this quadratic equation to get:


(x-17) * (x-13) = 0


this makes x = 17 or x = 13.


x can't be 13 because then one of the sides of the triangle will be 0.


x has to be 17.


when x is 17, you get:


x-12 = 17-12 = 5
x-13 = 17-13 = 4
x-14 = 17-14 = 3


that's your 345 triangle which is a right triangle.


side of the equilateral triangle is 17.