Three rods of equal lengths are joined to form an equilateral triangle ABC. D is the mid-point of AB. The coefficient of linear expansion is $\alpha _1$ for material of rod $AB$ and $\alpha _2$ for material of rods $AC$ and $BC$. If the distance $DC$ remains constant for small changes in temperature, then:

Question

$$\begin {array} {1 1} (1)\;\alpha_1= 2 \alpha _2 & \quad (2)\;\alpha_1=4\alpha _2 \\ (3)\;\alpha_1= 8 \alpha _2 & \quad (4)\;\alpha_1= \alpha _2 \end {array}$$