10 points and brainliest for correct answer!
a rectangle is inscribed in a square so that each vertex of the rectangle is located on one side of the square, and the sides of the rectangle are parallel to the diagonals of the square. suppose that one side of the rectangle is twice the length of the other and that the diagonal of the square is 12 meters long. find the sides of the rectangle.

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  The sides of the rectangle are 4 and 8 meters long.

Explanation:

Please consider the attached figure. Both the rectangle and the square are symmetrical about the x- and y-axes, so we only need to consider the corner of the rectangle in one quadrant.

If the rectangle is twice as long as wide, then the distance of the corner to one axis (we chose y) will be twice the distance to the other axis. The locus of points for which that is true is the line with slope 1/2. The corner of the rectangle will be on that line.

If the corner of the rectangle is also on the square, then it will be located at the point of intersection of the line with slope 1/2 and the line representing the edge of the square. That point of intersection is the point (4, 2) on this graph. That is, half the length of the rectangle is 4 m, and half the width is 2 m. This indicates the sides of the rectangle are 8 m and 4 m in length.

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The diagonals of the square are 12 m long, so orienting the square the way we have makes the x- and y-intercepts 6 units from the origin in each quadrant.