Question

which of the following is not true about variable stars?
they vary in brightness due to the expansion/contraction of their outer layers.
they are brightest at their smallest diameter.
certain variable stars have measurable relationships between brightness and pulsation periods
astronomers can find out how far away a star is based on its luminosity.

Explanation:

Variable stars vary in brightness due to expansion/contraction of outer layers. Cepheid variables have a period - luminosity relation. Astronomers use luminosity to find distance (standard candle method). However, variable stars are brightest when their outer layers expand (larger diameter), not at the smallest diameter.

Answer:

They are brightest at their smallest diameter.