The sides of a triangle have lengths x, x+4 and 30. Specify those values of x for which the triangle is acute with longest side 20

Question

I honestly have Bren struggling I'm geometry and K seriously need help 

Answer 1

x, x+4, and 20.  If the triangle is acute then x must be more than the value that would make the triangle into a right triangle.  For such a right triangle, if the longest side is 20, then that is the length of the hypotenuse.

 

x^2 + (x + 4)^2 = 20^2

x^2 + x^2 + 8x + 16 = 400

2x^2 + 8x - 384 = 0

 which we should simplify to

 

x^2 + 4x - 192 = 0

 

(x + 16) (x - 12) = 0

 

so x is either -16, or 12.  Since x is a length it cannot be negative,

so x = 12 yields a right triangle, and 

x > 12 gives an acute triangle.

 

That is a lower bound, the upper bound is imposed by the condition that (x + 4)< 20

x < 16

 

So, the range of values for x are 12 < x < 16