Question

A new light is going to be built at A(6,19) . The closest power line to a new light runs along the straight line containing points B(0,4) and C(12,10) . At what point should the lightning contractors connect to the power line, and how much cable will they need to reach the lamp? Each unit on the plan equals 1 m.

Round final answer to two decimal places if needed.
show working out please
analytic geometry

Answer 1

They would need 1.82 meters of cable, approximately.

Explanation:

According to the problem, the line segment BC is the closest power line. So, the lightning contractors should connect the new light A to the closest point of the power line, which is C, you can see this in the image attached.

To know the total length of cable they need, we must find the distance from point A to point C with the following formula

`d=\sqrt{(y_(2) -y_(1) )^(2) +(x_(2) -x_(1) )^(2) }`

Where
`A(6,19)` and
`C(12,10)`, replacing coordinates, we have

`d_(AC)=\sqrt{(10-19)^(2) +(12-6)^(2) }=\sqrt{(-9)^(2)+(6)^(2) } =√(81+36)=√(117)\\d_(AC) \approx 1.82 \ m`

Therefore, they would need 1.82 meters of cable, approximately.