Question

a parallelogram has side lengths of 13 and 17 and an angle that measures 64°. law of cosines: ( a^2 = b^2 + c^2 - 2bc cos(a) ) what is x, the length of the diagonal, to the nearest whole number? options: 18, 16, 19, ...

Explanation:

Step1: Identify values for Law of Cosines

In triangle \( WVY \), we have sides \( WV = 13 \), \( WY = 17 \), and angle \( \angle WVY = 64^\circ \). Let \( x = VY \). Using the Law of Cosines: \( x^2 = 13^2 + 17^2 - 2 \times 13 \times 17 \times \cos(64^\circ) \)

Step2: Calculate each term

First, calculate \( 13^2 = 169 \), \( 17^2 = 289 \). Then, \( 2 \times 13 \times 17 = 442 \). \( \cos(64^\circ) \approx 0.4384 \). So, \( 2 \times 13 \times 17 \times \cos(64^\circ) \approx 442 \times 0.4384 \approx 193.77 \)

Step3: Compute \( x^2 \)

\( x^2 = 169 + 289 - 193.77 = 458 - 193.77 = 264.23 \)

Step4: Find \( x \)

Take the square root: \( x = \sqrt{264.23} \approx 16.26 \), which rounds to 16.

Answer:

16