- Apply the Law of Cosines: \(a^2 = b^2 + c^2 - 2bc \cos(A)\).
- Substitute the given values \(b = 4\sqrt{2}\), \(c = 5\), and \(\angle A = 45^\circ\) into the formula.
- Calculate \(b^2 = 32\), \(c^2 = 25\), and \(2bc \cos(A) = 40\).
- Find \(a^2 = 32 + 25 - 40 = 17\), resulting in \(BC = \boxed{\sqrt{17}}\).
Explanation
- Identify given values and the Law of Cosines.
To find the length of side \(BC\) in triangle \(ABC\), we are given:
- The length of side \(AB\), denoted by \(c = 5\).
- The length of side \(AC\), denoted by \(b = 4\sqrt{2}\).
- The measure of angle \(A\) between sides \(AB\) and \(AC\), which is \(\angle A = 45^\circ\).
We can use the Law of Cosines to solve for the unknown side \(BC\), denoted by \(a\). The Law of Cosines states that in any triangle:
\[a^2 = b^2 + c^2 - 2bc \cos(A)\]
- Perform the necessary calculations.
Now, we substitute the known values into the Law of Cosines formula:
- \(b = 4\sqrt{2}\)
- \(c = 5\)
- \(\angle A = 45^\circ\)
First, calculate \(b^2\):
\[b^2 = (4\sqrt{2})^2 = 16 \times 2 = 32\]
Next, calculate \(c^2\):
\[c^2 = 5^2 = 25\]
Then, calculate the term \(2bc \cos(A)\):
\[2 \times (4\sqrt{2}) \times 5 \times \cos(45^\circ)\]
Since \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), we have:
\[2 \times 4\sqrt{2} \times 5 \times \frac{\sqrt{2}}{2} = 40 \times \frac{2}{2} = 40\]
- Determine the final length of side \(BC\).
Now, substitute these values back into the Law of Cosines equation to find \(a^2\):
\[a^2 = 32 + 25 - 40\]
\[a^2 = 57 - 40 = 17\]
Taking the square root of both sides gives the length of side \(BC\):
\[a = \sqrt{17}\]
Comparing this result to the given options, the correct length of side \(BC\) is \(\sqrt{17}\).
Examples
In navigation, if a pilot travels 5 miles in one direction and then turns 135 degrees to travel another \(4\sqrt{2}\) miles, the Law of Cosines can be used to calculate the direct distance back to the starting point. By understanding the relationship between two known sides and their included angle, we can accurately determine the third side of a triangular flight path, ensuring efficient route planning and fuel estimation.