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 and the included angle (60°), so we use the Law of Cosines. The Law of Cosines states that for a triangle with sides \( b \), \( c \), and included angle \( A \), \( a^2 = b^2 + c^2 - 2bc \cos A \). Here, \( b = 5 \), \( c = 21 \), and \( A = 60^\circ \), and \( \cos 60^\circ=\frac{1}{2} \).

Step2: Substitute Values into the Formula

Substitute \( b = 5 \), \( c = 21 \), and \( \cos 60^\circ=\frac{1}{2} \) into the Law of Cosines formula:

$$ a^2 = 5^2 + 21^2 - 2 \times 5 \times 21 \times \cos 60^\circ $$

$$ a^2 = 25 + 441 - 2 \times 5 \times 21 \times \frac{1}{2} $$

Step3: Simplify the Expression

First, simplify each term:

• \( 5^2 = 25 \), \( 21^2 = 441 \)
• The last term: \( 2 \times 5 \times 21 \times \frac{1}{2}= 5 \times 21 = 105 \)

Now, combine the terms:

$$ a^2 = 25 + 441 - 105 $$

$$ a^2 = 466 - 105 $$

$$ a^2 = 361 $$

Step4: Solve for \( a \)

Take the square root of both sides:

$$ a = \sqrt{361} = 19 $$

Answer:

\( 19 \)