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# help

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Help I see only one square here

Four of the points in the grid below form the vertices of a square \$S.\$  What is the sum of all the possible areas of \$S?\$  (Assume that neighboring points are one unit apart.)

Aug 16, 2023
edited by sandwich  Aug 16, 2023

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```
. . . . . . . .
. . . . . . . .
. . A . . . . .
. . . . . B . .
. . . . . . . .
. . . C . . . .
. . . . . . . .
```

In this grid, points A, B, and C form a square. Let's assume that each side of the square has a length of 1 unit. Now, we can calculate the areas of all the possible squares that can be formed by these points:

1. Square ABC: Since the side length is 1 unit, the area is 1 * 1 = 1 square unit.
2. Square ABDE (formed by extending the sides of ABC): The area is 2 * 2 = 4 square units.
3. Square AFGH (formed by extending the sides in another direction): The area is 2 * 2 = 4 square units.
4. Square AIJK (formed by extending the sides in both directions): The area is 3 * 3 = 9 square units.

The sum of all the possible areas is: 1 + 4 + 4 + 9 = 18 square units.

So, the sum of all the possible areas of squares that can be formed using the given points is 18 square units.

Aug 16, 2023