Two similar rods are joined as shown in figure. Then, temperature of junction is (assume no heat loss through lateral surface of rod and temperatures at the ends are shown in steady state)

Let $\theta$ be the temperature of junction.

H₁, H₂, H₃ and H₄ be the heat currents. Then,

H₁ = H₂ + H₃ + H₄

or   $\frac{100-\theta }{R}=\frac{\theta -50}{R}+\frac{\theta -50}{R}+\frac{\theta -0}{R}$

(where, R = thermal resistance of each part of two rods)
Solving the above equations, we get

$\mathit{\theta }={50}^{\circ }\mathrm{C}$