shortest distance between two points

Question

A house and a barn are on the same side of a driveway. Two walkways are planned to connect the buildings to a point P on the driveway. On a grid, the house would be at (5, 2), the barn would be at (-1, 4), and the driveway would be located along the x-axis. Where should point P be located to make the length of the walkways as short as possible?

Answer 1

Given: the point B(-1, 4) (the barn), the point H(5, 2) (the house). we need to find on the x-axis a point P(p, 0) such that sum of distances BP + PH will be the shortest.

Let reflect the point B across the x-axis. it will be the point B₁(-1, -4). For any point P on the x-axis BP = B₁P.

So if BP + PH will be the shortest, then B₁P + PH will be the shortest too.

Points B₁ and H are on different size of the x-axis. What is the shortest path from B₁ to H? Of course, a straight line segment!

So, to find the point P, you should connect points B₁(-1, -4) and H(5, 2) by the straight line and find the x-intercept of this line.

Hope, you can finish the rest yourself.

Answer 2

Well suppose P is at (x,0)

Then using the distance formula

Distance from house to P is √ (5-x)² + (2-0)² = f(x)

Distance from barn to P is

√ (-1-x)² + (4-0)² = g(x)

So total distance = f(x) + g(x)

Plot f(x) + g(x) on Desmos or a graphing calculator, and find the x value which gives the lowest y.

[ You should find x=3, y = 8.485]