Question

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

Explanation:

1. First, identify the values for the law - of - cosines formula:

• In the parallelogram, let \(b = 13\), \(c = 17\), and \(A=64^{\circ}\). We want to find the length of the diagonal \(x\) (using the law of cosines \(a^{2}=b^{2}+c^{2}-2bc\cos(A)\)).
• Substitute the values into the formula:
• \(b = 13\), \(c = 17\), and \(\cos(A)=\cos(64^{\circ})\approx0.4384\).
• \(x^{2}=13^{2}+17^{2}-2\times13\times17\times\cos(64^{\circ})\).
• Calculate \(13^{2}=169\), \(17^{2}=289\).
• Calculate \(2\times13\times17 = 442\).
• Then \(x^{2}=169 + 289-442\times0.4384\).
• \(x^{2}=169 + 289-193.8728\).
• \(x^{2}=458-193.8728\).
• \(x^{2}=264.1272\).
• Take the square - root of both sides: \(x=\sqrt{264.1272}\approx16.25\approx16\).

Answer:

16