Answer: \(a=330,\ b=-130\).

We start with the main method. Use the measurements and relationships shown in the diagram, then translate them into algebraic or trigonometric equations. Use the midpoint and distance formulae, keeping exact surd values where required.

The triangles are similar, so the scale factor from the smaller triangle to the larger triangle is

\(\frac{20}{4+2\sqrt5}.\)

The larger height is therefore

\((1+7\sqrt5)\frac{20}{4+2\sqrt5}.\)

First simplify the scale factor:

\(\frac{20}{4+2\sqrt5}=\frac{10}{2+\sqrt5}.\)

Rationalise the denominator:

\(\frac{10}{2+\sqrt5}\cdot\frac{2-\sqrt5}{2-\sqrt5} =\frac{10(2-\sqrt5)}{4-5}=10(\sqrt5-2).\)

So the larger height is

\((1+7\sqrt5)10(\sqrt5-2).\)

Now expand:

\((1+7\sqrt5)(\sqrt5-2)=\sqrt5-2+35-14\sqrt5=33-13\sqrt5.\)

Hence the larger height is

\(10(33-13\sqrt5)=330-130\sqrt5.\)

Therefore

\(a=330,\quad b=-130.\)

This completes the solution and gives the required result.