Is it possible to find an angle, given ratio of triangle size and length of one side?

Question

For this geometry:

...is it possible to find the angle (theta), given:

1. The value of $d$
2. The ratio $\frac{O_1}{O_2}$
3. The knowledge that the two angles are identical

I think this problem is underconstrained. How would one prove that is the case?

(FWIW, I know this is soluble given either the value of $O_1$ or $O_2$, but that's not what I'm looking for)

Thanks!

Edit:

Working through zoli's solution...

$$\frac{\sqrt{r^2 [(a+d)^2 +O_2^2] - a^2}}{a} = \frac{O_2}{a + d}$$

multiplying by $a$ and squaring

$$r^2 [(a+d)^2 +O_2^2] - a^2 = \left[\frac{a O_2}{a+d}\right]^2$$

expanding

$$r^2(a+d)^2 + r^2O_2^2= \frac{a^2}{(a+d)^2} O_2^2 + a^2$$

rearanging ...

$$O_2^2 \left(r^2 - \frac{a^2}{(a+d)^2} \right) = a^2 - r^2(a+d)^2$$

$$O_2 = \sqrt{\frac{a^2 - r^2(a+d)^2}{\left(r^2 - \frac{a^2}{(a+d)^2} \right)}}$$

On the surface, this looks like a plausible solution, but in testing it appears there's a problem somewhere (that I think may be just that the equations are not truly independent). Thinking more about this, I'm more convinced that the problem is underconstrained.

For example, take a 45 deg angle, $O_1 = 5, O_2 = 10 => d = 5; r = 0.5$

If I just scaled the whole thing down, though, the angle and $r$ would stay the same, but $d$ would change. And for any combo I choose, both the numerator and denominator end up as zero

It's totally possible it's a problem in my solving or transcription, though. I tested using the following JS code (paste into a chrome console with ctrl + shift + J)

function theta(r, d) {
 let a = (r*d) / (1-r);
 let num = a**2 - ( (r**2) * ((a+d)**2) );
 let denom = r**2 - (a**2 / ((a+d)**2));
 console.log(r, d, a, num, denom)
 return Math.sqrt(num/denom)
}

Answer 1

Since you say size of triangle, I assume the triangles are zoomed or scaled.The corrosponding sides are proportional. When placed similarly the corresponding sides are parallel.

Now for any angle $\theta$

$$ \dfrac{O_2}{O_1}= \frac {a+d}{a} =1 + \frac{d}{a} $$ Given are $O_1,O_2,d$; but not $a$, so the triangle cannot be solved.

and next since only two, not three informations are given for a triangle we cannot get $\theta$ by trig calculation.

Answer 2

For any $\theta, d, k = O_1/O_2$, extend the segment $d$ by $$x = \frac {k d} {1 - k}.$$ The ratio of the heights in the smaller and the larger right triangles will be $$\frac {h_1} {h_2} = \frac x {d + x} = k,$$ satisfying the constraints for arbitrary $\theta$.

Answer 3

Given d and the angle the ratio could not be determined (because the base of the triangle could be set to any value). A triangle can not be determined with only two length/angle informations. You need at least 3.

Therefore given d and the ratio, the angle is indeterminable as well.