To find the length of side \( a \) in triangle \( ABC \) using the Law of Sines, we start with the information provided:
- \( \angle C = 28^\circ \)
- Side \( c = 28 , \text{m} \)
According to the Law of Sines:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
This can be rearranged to find side \( a \):
\[ a = c \cdot \frac{\sin A}{\sin C} \]
However, we need to find \( \angle A \) in order to calculate \( a \). We can use the fact that the angles in a triangle sum up to \( 180^\circ \):
\[ A + B + C = 180^\circ \] \[ A + B + 28^\circ = 180^\circ \] \[ A + B = 152^\circ \]
Without additional information about angle \( B \), we can't directly solve this yet. Assuming \( A \) is unknown, we can express \( A \) in terms of \( B \):
\[ A = 152^\circ - B \]
We can still use the information about side \( c \) and \( \angle C \) to express \( a \):
Let’s estimate \( a \) for different values of \( A \) which implies different values for \( B \).
Since we can’t determine \( \angle A \) or \( \angle B \) without additional measurements or angles, let’s instead calculate an approximate value for \( a \) considering values of \( A \) that can be commonly found in certain right triangles.
To find out which of the provided answers \( a \) could be based on rough estimates and common angles, we can compare rough triangles but keep in mind the rule of sine to see which fits best.
We can calculate dimensions with assumed angles \( A \) (say using complementary values between \( 90^\circ \) and \( 28^\circ \)) or using common right triangles.
If \( A \approx 62 \) degrees, we have: Using \( \sin C = \sin(28^{\circ}) \) which is approximately \( 0.4695 \), thus \( \sin(62^{\circ}) \) gives us rough angles—this might correlate:
Performing the approximations and checks will lead us to values if we choose \( a \cdot 0.4695 = 28 \cdot 0.8746\) for rough sine values reciprocated.
Selecting angles leads back to:
Thus, if calculations were streamlined to approximation of relevant sine leading:
Assuming \( A = 62,\) appears typically close.
Checking answer estimates: So from options, if closest to side \( a \) yields a length approximation to prevalent triangles then.
The closest responsees in those parameters of calculation can expect to lead back roughly leading \( a \approx 53 m \):
So,
Final Answer: 53 m