Given that
A and B are two opposite points on Earth
Subtended central angle \theta = 1 \degree 54'
Diameter of the earth S = 1.276 * 10^{7} m
Distance between the earth and moon = d
Step 1: Create a figure based on the details provided.

Step 2: Calculating the distance between the earth and moon
Convert degrees minutes seconds to radians
\theta = 1 \degree 54' = 114' = 114* 60'' = 114 * 60 * 4.085* 10^{-6}
\theta = 3.32* 10^{-2} rad
From the figure
AB is a diameter of the Earth
Assumption: AB is considering as a arc.
Arc length S = \theta d
d = \frac{S}{\theta}
d = \frac{1.276 * 10^{7}}{3.32* 10^{-2}}
d = 3.84 * 10^8 m
Hence, distance between the earth and moon d = 3.84 * 10^8 m