Formula for the camping question
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Alice starts \(x_1\) miles west and \(y_1\) miles south of a campsite. Bob starts \(x_2\) miles east and \(y_2\) miles north of the campsite. They walk straight for the same distance, and meet exactly \(a\) miles east and \(b\) miles north of the campsite, coming from directions \(90^\circ\) apart. If \(a\) and \(b\) are both nonnegative, what is \(a\) and \(b?\)
Is there a formula for a and b?
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There is indeed a formula for \(a\) and \(b.\) However, this is general problem and there is no opportunity for any helpful tricks like there were in the Module 4 Day 13 Challenge problem.
To find the formula for \(a\) and \(b\) in this problem, you will have to use a geometric coordinate system. Here is an algorithm for how to do it:
Note: Let the point \(M\) be a midpoint of a segment \(\overline{AB}.\) The \(\triangle AB\text{(meeting point)}\) is a right isosceles triangle and \(M\) is a midpoint, so the distance from the point \(M\) to a meeting point (which is a median and height length of right isosceles \(\triangle\)) is equal to half of \(AB\) (which is the hypotenuse of the right isosceles \(\triangle\)).
Step 1: Find an equation of the line \(AB\) (where \(A\) and \(B\) are starting positions of Alice and Bob);
Step 2: Find the coordinates of point \(M;\)
Step 3: Find an equation of the line perpendicular to \(AB\) in point \(M\) using the equation of line \(AB\) and coordinates of point \(M;\)
Step 4: Find the length of segment \(\overline{AB};\)
Step 5: Measure out the length of a half of segment \(\overline{AB}\) on perpendicular to \(AB\) in point \(M\) and find the coordinates of the obtained point. This coordinate will give us our \(a\) and \(b.\)If you are interesting in finding the formula for \(a\) and \(b,\) you can try to do it yourself using this algorithm. However, it would be pretty complicated if you were not familiar to geometric coordinate system before.
So, once again, there exists a formula for \(a\) and \(b,\) but in this general problem there is no way to employ any helpful tricks in order to make it easier.