In triangle ABC, the length of side a is 5 centimeters, the length of side b is 8 centimet… — ACT

In triangle $ABC$ , the length of side $a$ is $5$ centimeters, the length of side $b$ is $8$ centimeters, and the measure of angle $C$ is $60^\circ$ . What is the length, in centimeters, of side $c$ ?

Answer: 7 cm

Answer

The length of side

c

7

centimeters.

Applying the Law of Cosines directly to the given Side-Angle-Side (SAS) triangle yields

c^2 = 5^2 + 8^2 - 2(5)(8) \cos 60^\circ = 25 + 64 - 40 = 49

, which gives

c = 7

Step-by-Step Solution

Identify the given values and the appropriate formula.

We are given two sides,

a = 5

and

b = 8

, and the included angle

C = 60^\circ

. To find the opposite side

c

, we use the Law of Cosines:

c^2 = a^2 + b^2 - 2ab \cos C

The Law of Cosines relates three sides of a triangle to the cosine of one of its angles, which is applicable for Side-Angle-Side (SAS) configurations.

Substitute the known values into the Law of Cosines equation.

c^2 = 5^2 + 8^2 - 2(5)(8) \cos 60^\circ

Plugging the given values into the formula allows us to solve for the unknown side

c

Evaluate the trigonometric and arithmetic terms.

Since

\cos 60^\circ = 0.5

, we get:

c^2 = 25 + 64 - 80(0.5)

c^2 = 89 - 40

c^2 = 49

Simplifying the expression step-by-step leads to the value of

c^2

Take the square root of both sides to find the side length.

c = \sqrt{49} = 7

Since side lengths must be positive, the square root of

49

gives the exact length of side

c

Key Concept

Law of Cosines

Question

Answer

Step-by-Step Solution

Key Concept

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