To find the length of side \( c \) using the Law of Sines, we start with the following relationship:
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
Given:
- \( A = 72^\circ \)
- \( a = 1246 , \text{mm} \)
We first need to determine angle \( C \). Since it's a right triangle, we know that the sum of angles in a triangle is \( 180^\circ \) and \( \angle B \) is \( 90^\circ \). Therefore:
\[ C = 90^\circ - A = 90^\circ - 72^\circ = 18^\circ \]
Now we can use the Law of Sines to find side \( c \):
- Calculate \( \sin A \) and \( \sin C \):
- \( \sin A = \sin(72^\circ) \)
- \( \sin C = \sin(18^\circ) \)
Using a calculator:
- \( \sin(72^\circ) \approx 0.9563 \)
- \( \sin(18^\circ) \approx 0.3090 \)
- Substitute the known values into the Law of Sines formula:
\[ \frac{1246}{0.9563} = \frac{c}{0.3090} \]
- Rearranging to solve for \( c \):
\[ c = \frac{1246 \cdot 0.3090}{0.9563} \]
- Calculate \( c \):
\[ c = \frac{385.154}{0.9563} \approx 403.5 , \text{mm} \]
Rounding to the nearest millimeter gives:
\[ c \approx 404 , \text{mm} \]
Considering the given options may contain slight rounding differences, the closest option is:
405 mm.