- Use the Law of Sines: \(\frac{BC}{\sin(A)} = \frac{AB}{\sin(C)}\).
- Substitute given values to find \(\sin(C) = \frac{\sqrt{3}}{2}\).
- Evaluate possible angles \(60^\circ\) and \(120^\circ\).
- Select \(C = 60^\circ\) based on the requirement that the triangle is acute.
- Final answer: \(\boxed{60^\circ}\).
Explanation
- Analyze the problem and identify the relevant geometric law.
We are given an acute triangle \(ABC\) where \(m(\angle A) = 45^\circ\), the side length \(AB = 10\sqrt{3}\), and the side length \(BC = 10\sqrt{2}\). We need to determine the measure of angle \(C\). Since the triangle is acute, all interior angles must be strictly less than \(90^\circ\). We use the Law of Sines, which states that the ratio of a side length to the sine of its opposite angle is constant for all sides of a triangle: \(\frac{BC}{\sin(A)} = \frac{AB}{\sin(C)}\).
- Apply the Law of Sines to find \(\sin(C)\).
Substitute the given values into the Law of Sines equation:
\(\frac{10\sqrt{2}}{\sin(45^\circ)} = \frac{10\sqrt{3}}{\sin(C)}\).
Knowing that \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\), we can substitute this value:
\(\frac{10\sqrt{2}}{\frac{\sqrt{2}}{2}} = \frac{10\sqrt{3}}{\sin(C)}\).
Simplifying the left side:
\(10\sqrt{2} \cdot \frac{2}{\sqrt{2}} = 20\).
So the equation becomes \(20 = \frac{10\sqrt{3}}{\sin(C)}\).
- Determine the angle measure and verify the conditions.
Rearrange the equation to solve for \(\sin(C)\):
\(\sin(C) = \frac{10\sqrt{3}}{20} = \frac{\sqrt{3}}{2}\).
The possible values for angle \(C\) such that \(\sin(C) = \frac{\sqrt{3}}{2}\) are \(C = 60^\circ\) or \(C = 120^\circ\). Given that the triangle \(ABC\) is acute, every angle must be less than \(90^\circ\). Thus, \(C = 120^\circ\) is excluded. Therefore, the only valid measure for angle \(C\) is \(60^\circ\). Checking the sum of the angles: \(m(\angle B) = 180^\circ - (45^\circ + 60^\circ) = 75^\circ\). Since all angles (\(45^\circ, 75^\circ, 60^\circ\)) are less than \(90^\circ\), this confirms that the triangle is indeed acute.
Examples
In architecture and civil engineering, understanding the properties of triangles is crucial for calculating structural stability. When designing a pitched roof or a support truss, knowing one side and its opposite angle allows engineers to determine other unknown dimensions or angles using the Law of Sines. For instance, if an architect knows the length of a base rafter \(BC = 10\sqrt{2}\) meters and the angle it makes with another support beam \(\angle A = 45^\circ\), they can calculate the angle \(\angle C\) that the structure makes with the ground, ensuring the stability and aesthetic requirements of the building are met.