Answer:
x=3
HI=18
Step-by-step explanation:
Equations
Suppose a segment with endpoints G, I. Somewhere in between lies point H, in such a way that:
GI = GH + HI
Every segment is given as an expression with variable x:
[tex]HI=6x[/tex]
[tex]GH=x^2+5x[/tex]
[tex]GI=7x+21[/tex]
Substituting:
[tex]7x+21=x^2+5x+6x[/tex]
Moving everything to the right side:
[tex]0=x^2+5x+6x-7x-21[/tex]
Swapping sides and simplifying:
[tex]x^2+4x-21=0[/tex]
Factoring:
[tex](x+7)(x-3)=0[/tex]
Solving, we have two possible solutions:
x=-7, x=3
The negative solution is discarded because it would result in negative lengths. Thus the solution is
x=3
Determine the length of HI:
HI=6x=6*3=18
HI=18
