On a square game board made up of small squares of side length 4 cm, the diagonal has a length of cm. Determine the number of small squares on the board.
If there are x squares on a side, each of length 4, then the diagonal is
d = 4x√2
so once you decide what d really is, just solve for x.
there are x^2 squares.
To determine the number of small squares on the board, we need to find the total number of squares that fit in both the length and width dimensions.
First, let's find the length of the diagonal using the Pythagorean theorem:
Diagonal^2 = Length^2 + Width^2
Substituting the given side length of the small squares (4 cm) into the equation, we have:
Diagonal^2 = 4^2 + 4^2
Diagonal^2 = 16 + 16
Diagonal^2 = 32
To find the diagonal, take the square root of both sides:
Diagonal = √(32)
Diagonal ≈ 5.657 cm
Since the diagonal of a square divides it into two congruent right-angled triangles, we can use the properties of these triangles to find the length of one side.
Let's label the length of one side of the square as "s". Using the Pythagorean theorem again for the right-angled triangle with sides labeled "s" and "s", we have:
Diagonal^2 = s^2 + s^2
32 = 2s^2
Dividing both sides by 2, we get:
16 = s^2
Taking the square root of both sides, we find:
s ≈ √(16) = 4 cm
This means that each side of the square is 4 cm long.
To find the number of small squares on the board, we need to determine how many squares fit both horizontally and vertically.
Let's divide the length of the board by the length of one side of the small square:
Total length of the board / Length of one side of the small square = Number of small squares in one row/column
Since the length of the board is equal to its width, and each side length of the small square is 4 cm, the equation becomes:
Length of one side of the board / Length of one side of the small square = Number of small squares in one row/column
Each side of the board is 4 cm long, so we have:
4 cm / 4 cm = 1
This means that there is 1 small square in each row and column.
To find the total number of small squares on the board, we multiply the number of small squares in one row and column together:
Number of small squares in one row/column * Number of small squares in one row/column = Total number of small squares on the board
1 * 1 = 1
Therefore, the number of small squares on the board is 1.
To determine the number of small squares on the board, we need to find out how many small squares fit along one side of the board and then multiply that by the number of squares along the other side.
The length of the diagonal of a square board can be found using the Pythagorean theorem, which states that for a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the hypotenuse (diagonal in this case).
In our case, the two shorter sides are the sides of a square, each measuring 4 cm. So, we have:
(4 cm)^2 + (4 cm)^2 = diagonal^2
16 cm^2 + 16 cm^2 = diagonal^2
32 cm^2 = diagonal^2
Taking the square root of both sides to solve for the length of the diagonal:
diagonal = √32 cm
diagonal ≈ 5.66 cm
Since the diagonal is also the hypotenuse of a right triangle formed by the sides of the square board, we know that the sides of the square board will have a length of approximately 5.66 cm.
To find the number of small squares along one side of the board, we can divide this length by the side length of each small square:
Number of small squares along one side = Length of diagonal / Side length of small square
Number of small squares along one side ≈ 5.66 cm / 4 cm
Number of small squares along one side ≈ 1.415 squares
Since a square grid must have whole numbers of squares along each side, we can round 1.415 up to the nearest whole number and assume that there are 2 squares along one side of the board.
To find the total number of small squares on the board, we multiply the number of squares along one side by the number of squares along the other side:
Total number of small squares = Number of small squares along one side^2
Total number of small squares ≈ 2^2
Total number of small squares ≈ 4 squares
Therefore, there are approximately 4 small squares on the board.