A ranger in a lookout tower spots two fires on the campground below. Fire A is 75 meters east and 40 meters south of the tower. Fire B is 37 meters west and 64 meters south of the tower.
If there is a fire hydrant located three-fourths of the way from Fire A to Fire B, find the location
of the fire hydrant relative to the tower.

The location of the hydrant relative to the tower is meters.

In this question we should use the vectorial formula for a line segment to determine the coordinates of the hydrant relative to the tower (), in meters:

(1)

Where:

Now we determine the location of the hydrant: (, , )

The location of the hydrant relative to the tower is meters.

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the fire hydrant is 9 meters west and 58 m south of the tower

Step-by-step explanation:

Fire A and Fire B can be represented as a point in the Cartesian coordinate with the tower as the origin. Fire A is 75 meters east  and 40 meters south of the tower, It can be represented as (75, -40).  Fire B is 37 meters west and 64 meters south of the tower,  It can be represented as (-37, -64).

If a point O(x, y) divides a line segment AB in the ratio n:m, with A() and B(), the point O(x, y) is at:

The fire hydrant (x, y) divides Fire A(75, -40) and fire B(-37, -64) in the ratio 3:1(three fourth), hence:

The fire hydrant is at (-9, -58). That is the fire hydrant is 9 meters west and 58 m south of the tower