Two straight paths diverge from a junction J at an angle of 120^\circ. A person walks 5\te… — JAMB UTME

Two straight paths diverge from a junction $J$ at an angle of $120^\circ$ . A person walks $5\text{ km}$ along the first path to point $A$ , and another person walks $16\text{ km}$ along the second path to point $B$ . What is the direct distance between $A$ and $B$ in kilometres?

Answer: 19 km

Answer

The direct distance between points

A

and

B

19\text{ km}

Using the Cosine Rule

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

with sides

5\text{ km}

and

16\text{ km}

and included angle

120^\circ

yields

c^2 = 5^2 + 16^2 - 2(5)(16)(-0.5) = 25 + 256 + 80 = 361

. Taking the square root gives

19\text{ km}

Step-by-Step Solution

Identify given values and setup the Cosine Rule formula

Side

a = 5

, side

b = 16

, and included angle

\theta = 120^\circ

The scenario provides two sides and the included angle (SAS configuration), which requires the Cosine Rule to find the third side.

Substitute the values into

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

c^2 = 5^2 + 16^2 - 2(5)(16) \cos(120^\circ)

Populating the formula allows evaluation of the unknown distance squared.

Evaluate the trigonometric term and simplify

c^2 = 25 + 256 - 160(-0.5) = 281 + 80 = 361

The cosine of an obtuse angle in the second quadrant (

120^\circ

) is negative:

\cos(120^\circ) = -0.5

Take the square root to find the distance

c

c = \sqrt{361} = 19\text{ km}

Taking the positive square root gives the physical distance between the two points.

Key Concept

Applying the Cosine Rule to find the length of an unknown side in a non-right triangle given two sides and the included angle (SAS).

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