Find the speed of of the third friend

Question

You and two friends want to meet up at a location equally distant from each of your homes. One of your friends lives 4 miles north of your home and the other lives 4 miles east. Determine the best place to meet such that you each travel the same distance to that point. A third friend will ride her bicycle to your meeting point from her home 4 miles northwest of your home. Assume that you and your first two friends walk at 4 miles per hour, what speed should the third friend ride to meet the group? Fully explain your answer in essay format to show your work. 

Answer 1

Let your and two friends’ homes be A, B, C - this will be a right triangle with right angle at A( your home), BC the hypotenuse. 

Let us graph these points in a graph paper. So the coordinates will be A(0,0), B( 0,4), C(4,0)

As per question if you draw the triangle you have AB=4mi, AC=4mi. Using Pythagorean theorem, BC= sqrt (4^2+4^2)= 4*sqrt(2)mi

the meeting point is equidistant from the homes, so it has to be the circumcenter of the triangle ABC. Let it be E. Since this is a right triangle, circumcenter, E, would be the midpoint of the hypotenuse and the distance from each of your homes would be half of the hypotenuse ( radius of the circle touching the vertices if the triangle). The coordinates of E(2,2)

Distance = BC/2= 2*sqrt(2) mi

Time taken = distance/speed = 2*sqrt(2)/4=

Sqrt(2)/2

Third friend lives in house D - imagine this us to the top left( northwest ) of A. AD = 4mi

To find the coordinates of D:

if you consider AD as the hypotenuse of a special right triangle ADF where F is a point on the Y axis, and it is a 45,45,90 special triangle, we know AD=x* sqrt(2)

4=x* sqrt(2)

x= 2sqrt(2)

The coordinates of D(-2sqrt(2),2sqrt(2))

nite that the x coordinate is negative because this friend lives north west.

Distance between DE can be calculated using distance formula. And you can calculate the speed of third friend using that distance and the time which you previously calculated.

Hope you can complete the last part.

Hope this helps! 

Answer 2

Draw and label a diagram!

You are at (0, 0), first friend at (0, 4), second friend at (4, 0)

The meeting point is (2, 2). Use distance formula to confirm.

The third friend is at (-√2, √2).

Use distance formula and rtd to answer second part.