Question

what is the approximate length of the third side of the triangle below? triangle with sides 31 mm, 27 mm, angle 75° between them, not drawn to scale options: 22.2 mm, 25.8 mm, 35.5 mm, 46.1 mm

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides (27 mm, 31 mm) and the included angle (75°). We use the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( a = 27 \), \( b = 31 \), \( C = 75^\circ \).

Step2: Calculate \( \cos(75^\circ) \)

\( \cos(75^\circ) \approx \cos(75) \approx 0.2588 \)

Step3: Substitute into the Formula

\( c^2 = 27^2 + 31^2 - 2\times27\times31\times0.2588 \)
\( 27^2 = 729 \), \( 31^2 = 961 \), \( 2\times27\times31 = 1674 \)
\( c^2 = 729 + 961 - 1674\times0.2588 \)
\( 729 + 961 = 1690 \)
\( 1674\times0.2588 \approx 433.2 \)
\( c^2 \approx 1690 - 433.2 = 1256.8 \)

Step4: Find \( c \)

\( c = \sqrt{1256.8} \approx 35.5 \)

Answer:

35.5 mm (corresponding to the option: 35.5 mm)