Given a square, what is the locus of points equidistant from the sides?
equidistant from all the sides? A point in the center. It is where the diagonals cross.
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The locus of points equidistant from the sides of a square is a set of points that form another square. This square is smaller and concentric, meaning it shares the same center as the original square.
To understand why this is the case, let's break it down step by step:
1. Consider a square with sides of length 'a'. Let's label the vertices of this square as A, B, C, and D.
2. Now, let's take an arbitrary point P that is equidistant from the sides of the square. This means that the distances from point P to each of the sides are equal.
3. To find the locus of points that fulfill this condition, we can consider the perpendicular bisectors of each of the sides. A perpendicular bisector is a line that passes through the midpoint of a segment at a right angle.
4. Since the square has equal sides, the perpendicular bisectors of each side will intersect at the center of the square. Let's label this point as O.
5. The intersection of these perpendicular bisectors forms the vertices of the smaller square that is concentric with the original square.
6. The distance from the center O to any of the vertices of the smaller square is half the length of one side of the original square, 'a/2'. Thus, the side length of the smaller square is 'a/2'.
In summary, the locus of points equidistant from the sides of a square is a smaller square that shares the same center as the original square, with its side length being half of the original square's side length.