Question

A rectangle with dimensions 60 units * 40 units is drawn on a grid paper that has a 1 unit * 1 unit grid. Each vertex of the rectangle is on a node, and each side of the rectangle is on a gridline. Then a diagonal of the rectangle is drawn. How many cells are cut into two(not necessarily equal) parts? PLEASE ANSWER ASAP!!!!!!!

Answer 1

To determine the number of cells cut by a diagonal in a 60 by 40 unit rectangle, calculate the GCD of the sides, use it to find a scaled down version of the grid, and then multiply the number of cuts in this smaller grid by the number of such blocks in the larger grid, resulting in a total of 1600 cut cells.

Step-by-step explanation:

To find out how many cells a diagonal cuts through in a rectangle of dimensions 60 units by 40 units on a grid paper, we need to consider the greatest common divisor (GCD) of the dimensions. The diagonal of a rectangle will intersect with horizontal and vertical lines within the grid. Each time the diagonal intersects a grid line, one cell will be cut into two parts. The number of cells cut will be the sum of horizontal and vertical grid lines intersected by the diagonal, minus 1 to avoid double counting the intersection that occurs at each corner.

First, we calculate the GCD of the two sides of the rectangle:

• GCD of 60 and 40 is 20.

As the diagonal crosses each of these lines, it cuts one rectangle in the grid in two. Therefore, the diagonal intersects:

• (60/20) + (40/20) - 1
• (3) + (2) - 1 = 4

So, the diagonal will cut through 4 cells. However, this does not give us the full number of cut cells, as it only accounts for those cut by horizontal and vertical intersections. Since both dimensions of the rectangle are divisible by the GCD, we can use a smaller rectangle with dimensions 3 by 2 to represent a scaled down version where the diagonal goes through every point a diagonal would in the larger rectangle. The number of cells the diagonal will cut in the larger grid will be the same as the number of cells it cuts in one block of this smaller grid, multiplied by the number of such blocks that fit into the larger grid.

The number of blocks that fit into the larger grid:

• (60/3) * (40/2) = 20 * 20 = 400

So, the diagonal cuts through 4 cells within each block and there are 400 such blocks. Therefore, the total number of cells cut into two by the diagonal in the entire grid is:

• 4 * 400 = 1600

The diagonal will cut a total of 1600 cells into two not necessarily equal parts in the given grid.