Four points form the vertices of a square. Find the area of the square if the points are (2,b), (0,a), (5,c), and (7,d).

We can solve this problem by recognizing opposite sides of a square have the same coordinates and using the distance formula.

 

Opposite Sides: Since the points form a square, opposite sides will have the same x-coordinate or the same y-coordinate.

 

Side Length: We can find the side length of the square by calculating the distance between two points with the same x-coordinate (or same y-coordinate).

 

Area of Square: Once we have the side length, the area of the square can be calculated using the formula: Area = side length squared.

 

Here's how to find the side length and area:

 

Side Length:

 

Since all points have different x-coordinates, we can focus on points with the same y-coordinate. For example, points (2, b) and (7, d) both have a y-coordinate of "b".

 

The distance between these two points represents the side length of the square. We can use the distance formula:

 

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

 

In this case:

 

Distance (side length) = √((7 - 2)^2 + (b - b)^2) (since y coordinates are the same, their difference is 0)

 

Distance = √((5)^2 + (0)^2)

 

Distance = √25 = 5 (We can simplify the radical)

 

Area of Square:

 

Now that we know the side length is 5, the area of the square can be calculated using the formula:

 

Area = side length ^ 2

 

Area = 5 ^ 2

 

Area = 25

 

Therefore, the area of the square is 25​ square units.