Find coordinates of point A with identical rectangles

In this problem, we have three identical rectangles placed on a coordinate grid, and we need to find the coordinates of point A.

Let's define the dimensions of these identical rectangles. We'll call the long side 'L' and the short side 'w'.

Let's trace the horizontal distance from the point at three comma two to the point at fifteen comma seven. Looking at the x-coordinates, the total horizontal distance is fifteen minus three, which equals twelve.

Visually, this horizontal distance is made up of one length plus one width plus another length from the three rectangles. However, let's look closer at the layout.

Let's set up equations based on the coordinate gaps. The total change in x from the bottom left corner to the top right corner is twelve units.

Wait, let's re-examine the diagram carefully. The horizontal span from x equals three to x equals fifteen is comprised of one length of the first rectangle, then the width of the vertical rectangle, then the length of the third rectangle. So, two L plus w equals twelve.

Now let's look at the vertical change. The y-coordinate goes from two to seven, so the total height difference is five.

Looking at the vertical path, we have the height of the first rectangle, which is w, then we move up to the top of the middle rectangle, then down to the start of the last rectangle. Effectively, the height difference is one length minus one width.

We now have a system of two equations. Two L plus w equals twelve, and L minus w equals five.