Find Midpoint of Shaped Area: 300, 5000, 1500, 5142

[iNCREDiBLE]

I need to find the mid point of the area of the shape described below. This would divide the area into 4 subparts of equal area.. The dimensions of the shape are: north side is 300 feet, east side is 5000 feet, south side is 1500 feet, west side is 5142 feet. The south side is perpendicular to the east side.

 

[AKG]

Draw a picture of it first. Now pick some arbitrary point on the east side, and draw a horizontal line there. Also draw a horizontal line at the lower of the two top points. You'll end up with a triangle on top of a trapezoid on top of another trapezoid. Let's say that the point you picked arbitrarily on the east side is x feet above the south side. Then you should be able to get an expression in terms of x for the area of the trapezoid below, and the trapezoid above. The area of the triangle will remain fixed. You want to solve for x such that the area of the bottom trapezoid is equal to the sum of the areas of the top trapezoid and the top triangle. Of course, you'll need to apply some basic trigonometry and geometry to find these areas in terms of x, and some basic algebra to solve for x. Do a similar thing picking an arbitrary point y on the south side, so the area to the left is equal to the area to the right. Upon picking an arbitrary point, you might find the shapes to the left and right are weird, so add lines so that you can break them up into simple shapes like trapezoids and triangles. Compute the left and right areas in terms of y and solve for y such that the areas are equal. Then the line at horizontal line at x and the vertical line at y will meet at the centroid.

 

[thepatientmental]

To find the midpoint of the shaped area, we first need to calculate the total area of the shape. This can be done by multiplying the length of the north and south sides, and then multiplying the length of the east and west sides. In this case, the total area would be 300 x 1500 = 450,000 square feet.

Next, we need to find the midpoint of each side. The midpoint of a line segment can be found by taking the average of the two endpoints. So for the north side, the midpoint would be (0 + 300)/2 = 150 feet. For the east side, the midpoint would be (0 + 5000)/2 = 2500 feet. For the south side, the midpoint would be (0 + 1500)/2 = 750 feet. And for the west side, the midpoint would be (0 + 5142)/2 = 2571 feet.

Now that we have the midpoints of each side, we can draw a line connecting them to create a cross in the center of the shape. This cross will divide the shape into 4 equal parts, with each part having an area of 112,500 square feet (450,000/4).

To find the exact midpoint of the shaped area, we can take the average of the midpoints of the north and south sides, and the average of the midpoints of the east and west sides. So the midpoint of the north and south sides would be (150 + 750)/2 = 450 feet, and the midpoint of the east and west sides would be (2500 + 2571)/2 = 2535.5 feet.

Therefore, the midpoint of the shaped area would be located at (450, 2535.5). This point would divide the area into 4 equal subparts, with each subpart having an area of 112,500 square feet.