Two forest rangers are standing on a road that runs east-west. One sights a forest fire in the direction 31° east of north. The other sights the same fire 57° west of north.
Find the distance from each ranger to the forest fire and
the shortest distance from the fire to the road.

The forest ranger closest to the fire is at a distance of 9.92 miles (nearest hundredth).

Draw a horizontal line AB of length 15 miles to represent the positions of the two forest rangers.

Point A is to the left and point B is to the right.

Complete triangle ABC so that m∠A = 41° and m∠B = 56°.

The point C represents the location of the fire.

Let x = length of BC. It is the shortest distance from the forest ranger B to the fire because BC is opposite the smaller angle of 41°.

Because the sum of angles in a triangle is 180°,

m∠C = 180° - (41° + 56°) = 83°.

From the Law of Sines,

x/sin(41) = 15/sin(83)

x = 15*[sin(41)/sin(83)] = 9.915 mi

Step-by-step explanation:

Draw a horizontal line AB of length 15 miles to represent the positions of the two forest rangers.

Point A is to the left and point B is to the right.

Complete triangle ABC so that m∠A = 41° and m∠B = 56°.

The point C represents the location of the fire.

Let x = length of BC. It is the shortest distance from the forest ranger B to the fire because BC is opposite the smaller angle of 41°.

Because the sum of angles in a triangle is 180°,

m∠C = 180° - (41° + 56°) = 83°.

From the Law of Sines,

x/sin(41) = 15/sin(83)

x = 15*[sin(41)/sin(83)] = 9.915 mi

Step-by-step explanation:

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