Question

5.
find the area and side length of the square.
area =
side length =

Explanation:

Step1: Assume grid square side is 1 unit.

Looking at the square (diamond - shaped) on the grid, we can use the Pythagorean theorem to find the side length. Let's say from one vertex to the adjacent vertex, the horizontal and vertical distances are 3 and 4 units (by counting grid squares). So, for a side of the square, using \(a = 3\), \(b = 4\), the side length \(s=\sqrt{3^{2}+4^{2}}\).

$$s=\sqrt{9 + 16}=\sqrt{25}=5$$

Step2: Calculate the area of the square.

The formula for the area of a square is \(A=s^{2}\). Since \(s = 5\), then \(A = 5^{2}=25\) square units. (Alternatively, we can use the method of subtracting the area of the triangles from the area of the enclosing rectangle. The enclosing rectangle would have length \(8\) and width \(8\)? Wait, no, actually, looking at the grid, the square is formed such that the enclosing rectangle has length \(7\)? Wait, no, my first method: when we have a square rotated, the side is the hypotenuse of a right triangle with legs 3 and 4 (counting grid squares: from the left - most point to the next, horizontal 3, vertical 4). So side length is 5, area is \(5\times5 = 25\). Another way: the square can be thought of as having diagonals? Wait, no, the first method is correct. Let's re - check: if we count the grid, the horizontal distance between two adjacent vertices is 3, vertical is 4, so by Pythagoras, side is 5. Then area is \(5^{2}=25\).

Answer:

Area = 25 square units, Side length = 5 units