In a triangle with a \( 45^{\circ} \) angle and sides of length 5 m and 8 m , what is the measure of the the third side? 3 m m m It cannot be determined from the information provided.

To find the length of the third side of a triangle with a \( 45^\circ \) angle and two sides of lengths 5 m and 8 m, we can use the Law of Cosines. The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] where \( c \) is the side opposite angle \( C \), and \( a \) and \( b \) are the other two sides. Let's denote: - \( a = 5 \) m - \( b = 8 \) m - \( C = 45^\circ \) Now we can substitute these values into the formula: \[ c^2 = 5^2 + 8^2 - 2 \cdot 5 \cdot 8 \cdot \cos(45^\circ) \] We know that \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \). Now we can calculate: \[ c^2 = 25 + 64 - 2 \cdot 5 \cdot 8 \cdot \frac{\sqrt{2}}{2} \] \[ c^2 = 25 + 64 - 40\sqrt{2} \] \[ c^2 = 89 - 40\sqrt{2} \] Now we can find \( c \): \[ c = \sqrt{89 - 40\sqrt{2}} \] To get a numerical approximation, we can calculate \( 40\sqrt{2} \): \[ \sqrt{2} \approx 1.414 \implies 40\sqrt{2} \approx 40 \cdot 1.414 \approx 56.56 \] Now substituting this back into the equation: \[ c^2 \approx 89 - 56.56 \approx 32.44 \] \[ c \approx \sqrt{32.44} \approx 5.7 \text{ m} \] Thus, the length of the third side is approximately \( 5.7 \) m.