The area of a square is 10 square meter. what is the product of the lengths of the diagonals of the square?
let each side be x
x^2 = 10
x=√10
diagonal ^2 = √10^2+√10^2 = 20
diagonal = √20
product of its diagonal = √20√20 = 20
To find the product of the lengths of the diagonals of a square, we'll need to know the length of its sides.
Let's start by finding the length of one side of the square. Since the area of the square is given as 10 square meters, we can use the formula for the area of a square:
Area = side length * side length
Substituting the given area of 10 square meters into the formula, we get:
10 = side length * side length
To solve for the side length, we can take the square root of both sides of the equation:
√10 = side length
Now that we have the length of one side of the square, we can calculate the lengths of the diagonals. In a square, the diagonals are equal in length and form right angles with each other.
To calculate the length of a diagonal, we can use the Pythagorean theorem:
Diagonal length = √(side length^2 + side length^2)
Since the lengths of the diagonals are equal in a square, we can find the product of their lengths by squaring the length of one diagonal:
Product of diagonal lengths = (length of one diagonal)^2
Let's calculate the length of one diagonal:
Diagonal length = √(side length^2 + side length^2)
= √(2 * side length^2)
= √(2 * (√10)^2) (substituting √10 for side length)
= √(2 * 10)
= √20
Now we can find the product of the diagonal lengths:
Product of diagonal lengths = (√20)^2
= 20
Therefore, the product of the lengths of the diagonals of the square is 20.