# Find the center of mass of the rectangle `(1,-3)(4,-3)(4,5)(1,5)`

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### 2 Answers

The center of mass of the rectangle with vertexes (1, -3), (4, -3), (4, 5) and (1, 5) has to be determined. Assuming the material that the rectangle is made of has a uniform density, the center of mass of the rectangle lies at the intersection of its diagonals.

This is the intersection of the liens between (1, -3) and (4, 5) and between (4, -3) and (1, 5)

The line between (1, -3) and (4, 5) is (y +3)/(x - 1) = (8/3)

=> 3y + 9 = 8x - 8

=> y = (8x -17)/3

The line between (4, -3) and (1, 5) is (y +3)/(x - 4) = (8/-3)

=> -3y - 9 = 8x - 32

=> y = (8x - 23)/(-3)

(8x -17)/3 = (8x - 23)/(-3)

=> 8x - 17 = 23 - 8x

=> 16x = 40

=> x = 2.5

y = 1

**The center of mass is (2.5, 1)**

An easier way to determine the center of mass of the rectangle with vertices (1, -3), (4, -3), (4, 5) and (1, 5) is to look at the coordinates of the points.

The center of mass is the point equidistant from the 4 points.

The points (1, -3) and (4, -3) have a common y-coordinate, -3. The points (4, 5) and (1, 5) have a common y-coordinate 5. The y-coordinate of the point equidistant from the points is equal to (-3 + 5)/2 = 2/2 = 1

Similarly, the points (4, -3) and (4, 5) have a common x-coordinate 4 and the points (1, -3) and (1, 5) have a common x-coordinate 1. The x-coordinate of the equidistant point is (1 + 4)/2 = 2.5

This gives the coordinates of the center of mass as (2.5, 1)