2023 AMC 10B Problems/Problem 24
What is the perimeter of the boundary of the region consisting of all points which can be expressed as with , and ?
Solution 1
Notice that this we are given a parametric form of the region, and is used in both and . We first fix and to , and graph from :
Now, when we vary from to , this line is translated to the right units:
We know that any points in the region between the line (or rather segment) and its translation satisfy and , so we shade in the region:
We can also shift this quadrilateral one unit up, because of . Thus, this is our figure:
The length of the boundary is simply ( can be obtained by Pythagorean theorem, since we have side lengths and .). This equals
~Technodoggo