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.

The answer is 4√3 MPH. Here's why:

(I refer to North friend as Bob, West friend as Joe, and Northeast friend as Kelly. I also refer to locations as 'points'.  

Another note, just to clarify: To divide 2 radicals (Ex: 6√4 / 2√2), you need to divide the numbers outside the radical symbol by each other, and then divide the numbers inside the radical symbol by each other.  

(6√4 / 2√2 would equal 3√2, because 6 divided by 2 is 3, and 4 divided by 2 is 2.  

1. Ignoring Kelly, for now, we need to find a point that is of equal distance away from the other two friends. We can do this by drawing a square, using the 3 points, with side lengths of 4, since we know that they live 4 miles apart. The middle of this square is the place they have to meet since it is of equal distance from each point

2.  Now we need to find the distance from Bob to Joe, (aka the slope). Pythagorean's Therom says that for a right triangle, (height)²+(base)²=(hypotenuse)². Therefore, (4)²+(4)²=(x)², where x is the hypotenuse. This can be simplified to 32=x². Therefore, the slope is √32. When we simplify the radical, it becomes 4√2. However now, we need to divide this by 2, because each friend is only going halfway across the slope. This means that each friend will travel 2√2 miles.  

3. We know the Time(T)=Distance(D)/Speed(S). Therefore, the time for each friend to reach the location will be 2√2/4. The speed is 4 because the friends go 4 MPH. This information will come into play later on.  

4. Now we need to find out the MPH Kelly should go to reach that same point as the rest of her friends. First, however, we need to find the distance from her point to the meeting point. We can use Pythagorean's Therom again because the meeting point, my point, and Kelly's point all form a right triangle. Again, (height)²+ (base)²= (hypotenuse)². Therefore, (4)²+(2√2)²=y², where y is the distance from Kelly to the meeting point. This simplifies to (16) + (8) = (y²).  16+8=24, therefore y is √24 miles. We can simplify this to 2√6. Now we know that the distance from Kelly to the center point is 2√6 miles, we can find the speed.  

5. As already stated, T=D/S. In order for all friends to meet at the same time,  the amount of time it takes for Group 1 to reach the center point must be equivalent to Kelly's.  

We learned earlier that 2√2/ 4= T. This means that 2√2 / 4 = 2√6 / z. This is because 2√6 was the distance Kelly lived away from the center point, and we know that since the time will be the same, the fractions will be the same.  

6. Now we can cross simplify: (4)(2√6)=(2√2)(z). This reduces to              (8√6)=(2√2)(z). When we divide each side by 2√2, we get our answer, 4√3

Therefore, Kelly needs to ride 4√3 MPH on her bike in order to reach her friends at the same time. Hope this helps :)

Step-by-step explanation:

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