Question

find the length of side ( x ). round to the nearest tenth.

( x = )
enter your answer as an integer or decimal number. examples: 3, -4, 5.5172
que enter dne for does not exist, oo for infinity
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Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In this triangle, let \(a = 10\), \(b = 12\), \(\theta=40^{\circ}\), and \(c=x\). Then \(x^{2}=10^{2}+12^{2}-2\times10\times12\times\cos40^{\circ}\).

Step2: Calculate each term

First, \(10^{2}=100\), \(12^{2}=144\). Then, \(\cos40^{\circ}\approx0.7660\), and \(2\times10\times12\times\cos40^{\circ}=240\times0.7660 = 183.84\).

Step3: Compute \(x^{2}\)

\(x^{2}=100 + 144-183.84=60.16\).

Step4: Find \(x\)

Take the square - root of \(x^{2}\), \(x=\sqrt{60.16}\approx7.8\).

Answer:

\(7.8\)