Thermal Expansion of a Triangular Frame

[SilverSoldier]

Homework Statement

$ABC$ is an equilateral triangle constructed out of 5 rods, $AD$ and $DB$ having coefficient of expansion $\alpha$, $BC$ and $CA$ having coefficient of expansion $\lambda$ and $CD$ having coefficient of expansion $\mu$.
Find in terms of $\alpha$ and $\lambda$ a value for $\mu$ so that rod $CD$ shan't break, when the temperature of the system is increased by $\theta$, given $\alpha^2\theta^2$, $\lambda^2\theta^2$ and $\mu^2\theta^2$ are negligible.

Relevant Equations

$e = l\alpha\theta$, where $e$ = expansion, $l$ = initial length, $\alpha$ = coefficient of expansion, and $\theta$ = change in temperature

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Suppose each side has initial length $l$. The solution taught to me is as follows.

Considering the lengths of the rods after expansion, we write $$\dfrac{3l^2}{4}\left(1+\mu\theta\right)^2+\dfrac{l^2}{4}\left(1+\alpha\theta\right)^2=l^2\left(1+\lambda\theta\right)^2$$ according to the Pythagorean Theorem, which yields $$\mu=\dfrac{4\lambda-\alpha}{3}.$$ My question is, is there any reason to assume that the angle at $D$ remains a right angle?

The way I understand it, due to symmetry, any horizontal movement of points $C$ and $D$ is prevented by equal and opposite horizontal forces caused at $C$ and $D$, so $CD$ may only expand vertically, but no such statement could be made about points $A$ and $B$. So, isn't it possible that it expands into a structure like this?

Is it correct to say that the situation described in the solution happens only if $D$ were clamped to a fixed point, or held fixed by some other means?

 

[haruspex]

is there any reason to assume that the angle at D remains a right angle?

If the angles of the triangles are permitted to change then there is no reason for anything to break, so you have to assume that any change in an angle constitutes breakage.

 

[kuruman]

Suppose that, upon heating, the original triangle expands in a way that it retains its equilateral shape (much like enlarging a photograph). Then all angles will remain the same as @haruspex noted. At the bottom left corner, you will have something like the blue triangle shown below. Any ideas where to go from here?