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The exercise has two square roots and some extra stuff.

To solve by hand, one would

isolate one square root on one side of the equation, to get something like a = b + c where a and b are square roots and c is not
square both sides (this can lose information so we'll have to check, among all solutions of this new equation, which ones are solutions of the original equation)
this gets rid of one of the square roots, the other one survives in the double product of (b + c)^2, as (2bc)
isolate the square root one one side of the equation
square both sides again
finally solve the equation with no square roots (a polynomial equation of degree four)
check which solutions satisfy the original equation

We can follow this method with Sage too, guiding it through the calculations.

It's a bit tedious but it saves us from doing the computations by hand.

Set constants:

sage: n = 4/3
sage: h1 = 1
sage: h2 = 1

Two variables in the symbolic ring and an equation:

sage: x, y = SR.var('x, y')
sage: g = sqrt(x^2 + y^2) + n * sqrt(x^2 + (h1 + h2 - y)^2) - (h1 + n * h2) == 0
sage: g
4/3*sqrt(x^2 + (y - 2)^2) + sqrt(x^2 + y^2) - 7/3 == 0

Solve does not help much:

sage: solve(g.subs({x: 1/5}), y)
[sqrt(25*y^2 - 100*y + 101) == -3/4*sqrt(25*y^2 + 1) + 35/4]

Substitute:

sage: h = g.subs({x: 1/5})
sage: h
4/15*sqrt(25*(y - 2)^2 + 1) + 1/5*sqrt(25*y^2 + 1) - 7/3 == 0

Get rid of denominators:

sage: hh = 15 * h
sage: hh
4*sqrt(25*(y - 2)^2 + 1) + 3*sqrt(25*y^2 + 1) - 35 == 0

Give names to the three summands:

sage: a, b, c = hh.lhs().operands()
sage: a, b, c
(4*sqrt(25*(y - 2)^2 + 1), 3*sqrt(25*y^2 + 1), -35)

Isolate one and square both sides:

sage: j = a^2 == (b + c)^2
sage: j
400*(y - 2)^2 + 16 == (3*sqrt(25*y^2 + 1) - 35)^2

Expand:

sage: jj = j.expand()
sage: jj
400*y^2 - 1600*y + 1616 == 225*y^2 - 210*sqrt(25*y^2 + 1) + 1234

Compute the difference of both sides (that should be zero):

sage: k = jj.lhs() - jj.rhs()
sage: k
175*y^2 - 1600*y + 210*sqrt(25*y^2 + 1) + 382

Give names to the four summands:

sage: aa, bb, cc, dd = k.operands()
sage: aa, bb, cc, dd
(175*y^2, -1600*y, 210*sqrt(25*y^2 + 1), 382)

If the sum is zero, then the difference of squares must be zero:

sage: m = (aa + bb + dd)^2 - (cc)^2
sage: m
(175*y^2 - 1600*y + 382)^2 - 1102500*y^2 - 44100

Expand:

sage: mm = m.expand()
sage: mm
30625*y^4 - 560000*y^3 + 1591200*y^2 - 1222400*y + 101824

Move from the symbolic ring to a proper polynomial ring:

sage: R.<y> = QQ[]
sage: p = R(mm)
sage: p
30625*y^4 - 560000*y^3 + 1591200*y^2 - 1222400*y + 101824

Find the roots as algebraic numbers:

sage: rr = p.roots(QQbar, multiplicities=False)
sage: rr
[0.09455004564983336?,
 1.144286167075680?,
 2.049032178063518?,
 14.99784589492526?]

Substitute in the original equation to know which give zero:

sage: ss = [h.subs({y: r}).lhs() for r in rr]
sage: ss
[4/15*sqrt(91.76848821332630?) + 1/5*sqrt(1.223492778309640?) - 7/3,
 1/5*sqrt(33.73477080401876?) + 4/15*sqrt(19.30615409645079?) - 7/3,
 1/5*sqrt(105.9633216684931?) + 4/15*sqrt(1.060103862141313?) - 7/3,
 1/5*sqrt(5624.384537198158?) + 4/15*sqrt(4224.599947705633?) - 7/3]

The symbolic ring keeps things too symbolic, move to algebraic numbers:

sage: tt = [QQbar(s) for s in ss]
sage: tt
[0.4424464312541602?, 0.?e-16, 0.?e-16, 29.99835872096514?]

Exactify to make sure things that look like zero are actually zero:

sage: _ = [t.exactify() for t in tt]
sage: tt
[0.4424464312541602?, 0, 0, 29.99835872096514?]

Select the roots of the transformed equation that are actually roots of the original equation:

sage: zz = [r for i, r in enumerate(rr) if tt[i].is_zero()]
sage: zz
[1.144286167075680?, 2.049032178063518?]

These are algebraic numbers:

sage: za, zb = zz
sage: za.minpoly()
x^4 - 128/7*x^3 + 63648/1225*x^2 - 48896/1225*x + 101824/30625
sage: zb.minpoly()
x^4 - 128/7*x^3 + 63648/1225*x^2 - 48896/1225*x + 101824/30625

Since their minimal polynomial is of degree four, they can be expressed with radicals, although the radical expression is not particularly illuminating.

sage: za.radical_expression()
-1/2450*sqrt((1500625*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(2/3) + 73460800*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) + 91681024)/(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3)) + 1/2*sqrt(-(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) - 91681024/1500625/(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) - 806400/sqrt((1500625*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(2/3) + 73460800*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) + 91681024)/(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3)) + 119936/1225) + 32/7
sage: zb.radical_expression()
1/2450*sqrt((1500625*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(2/3) + 73460800*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) + 91681024)/(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3)) - 1/2*sqrt(-(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) - 91681024/1500625/(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) + 806400/sqrt((1500625*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(2/3) + 73460800*(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3) + 91681024)/(12288/42875*I*sqrt(2776173) - 6128672768/1838265625)^(1/3)) + 119936/1225) + 32/7