Question

find the exact length of the unknown side of the triangle shown to the right. do not use a calculator. a = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides (7 and 15) and the included angle (60°). So we use the Law of Cosines. The Law of Cosines formula for a side \( a \) opposite angle \( A \) (but here we have sides \( b = 7 \), \( c = 15 \), and included angle \( C = 60^\circ \)) is \( a^{2}=b^{2}+c^{2}-2bc\cos(C) \).

Step2: Substitute the Values

Substitute \( b = 7 \), \( c = 15 \), and \( \cos(60^\circ)=\frac{1}{2} \) into the formula:

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Step3: Solve for \( a \)

Take the square root of both sides: \( a=\sqrt{169} = 13 \) (since length is positive).

Answer:

13