In \Delta ABC, side a = 6\text{ cm}, side b = 10\text{ cm}, and the included angle \angle … — JAMB UTME

In $\Delta ABC$ , side $a = 6\text{ cm}$ , side $b = 10\text{ cm}$ , and the included angle $\angle C = 120^\circ$ . What is the length of side $c$ ?

The length of side

c

14\text{ cm}

According to the Cosine Rule

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

, substituting the given values

a = 6

b = 10

, and

\cos 120^\circ = -\frac{1}{2}

yields

c^2 = 36 + 100 - 2(6)(10)\left(-\frac{1}{2}\right) = 136 + 60 = 196

. Taking the positive square root gives

c = 14\text{ cm}

Identify the given values and state the relevant Cosine Rule formula

Given:

a = 6\text{ cm}

b = 10\text{ cm}

\angle C = 120^\circ

. Formula:

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

Since two sides and the included angle (SAS configuration) are known, the Cosine Rule must be used to find the third side.

Evaluate

\cos 120^\circ

and substitute all values into the formula

\cos 120^\circ = -\frac{1}{2}

. Thus,

c^2 = 6^2 + 10^2 - 2(6)(10)\left(-\frac{1}{2}\right)

Cosine of an obtuse angle in the second quadrant is negative.

Simplify the algebraic expression

c^2 = 36 + 100 + 60 = 196

Multiplying

-2(60)\left(-\frac{1}{2}\right)

yields

+60

Take the principal square root to solve for

c

c = \sqrt{196} = 14\text{ cm}

Length must be a positive real number.

Cosine Rule for finding an unknown side in SAS triangle configurations

Question