Length of an edge

My sister asked me to help her with her homework for mathematics, however frustratingly I was not able to figure out how to solve it.

The assignment is as follows where it was requested to calculate the length between G and I. How should this assignment be solved?

Thanks!

• Please include your sister's effort as well as your attempts to solve it. – John Douma Jan 25 at 17:30
• What has this to do with Pythagorean-triples? – José Carlos Santos Jan 25 at 17:31
• As general advice, always compute what you can compute. Here, $\overline {HE}$ is easy. From that you can get $\overline {GE}$. Now you've got two right triangles left that you haven't used... – lulu Jan 25 at 17:34

It is a messy plane geometry and algebra problem. All the following using Pythagorean theorem. First $$|H-E|=4$$, then $$|E-G|^2=32$$. Let $$x=|E-I|$$ and $$h=|G-I|$$. The equation for $$x$$ and $$h$$ are $$x^2+h^2=32$$ and $$(5-x)^2+h^2=49$$. Solving these equations gives $$x=.8$$ and $$h=5.6$$, where $$h$$ is the answer to the problem.
Hint: Let $$\angle{EFG}=\alpha$$ then $$\cos(\alpha)=\frac{3}{5}$$ and $$\sin(\alpha)=\frac{GI}{7}$$