  Low radio frequency surveys are important for testing unified models of
radio-loud quasars and radio galaxies. Intrinsically similar sources that are
randomly oriented on the sky will have different projected linear sizes.
Measuring the projected linear sizes of these sources provides an indication of
their orientation. Steep-spectrum isotropic radio emission allows for
orientation-free sample selection at low radio frequencies. We use a new radio
survey of the Bo\"otes field at 150 MHz made with the Low Frequency Array
(LOFAR) to select a sample of radio sources. We identify 44 radio galaxies and
16 quasars with powers $P>10^{25.5}$ W Hz$^{-1}$ at 150 MHz using cross-matched
multi-wavelength information from the AGN and Galaxy Evolution Survey (AGES),
which provides spectroscopic redshifts. We find that LOFAR-detected radio
sources with steep spectra have projected linear sizes that are on average
4.4$\pm$1.4 larger than those with flat spectra. The projected linear sizes of
radio galaxies are on average 3.1$\pm$1.0 larger than those of quasars
(2.0$\pm$0.3 after correcting for redshift evolution). Combining these results
with three previous surveys, we find that the projected linear sizes of radio
galaxies and quasars depend on redshift but not on power. The projected linear
size ratio does not correlate with either parameter. The LOFAR data is
consistent within the uncertainties with theoretical predictions of the
correlation between the quasar fraction and linear size ratio, based on an
orientation-based unification scheme.
