Find the area of a square with a diagonal of 8 cm.
let each side be x
by Pythagoras:
x^2 + x^2 = 8^2
2x^2 = 64
x^2 = 32
since x^2 is also the area, the area is 32 cm^2
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Use the Pythagorean theorem:
x^2 + x^2 = 8^2
Solve for x.
To find the area of a square with a diagonal, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In the case of a square, the diagonal is the hypotenuse, and the sides are equal in length.
Let's call the length of each side of the square "s". We can form a right triangle by drawing a diagonal across the square, which will split the square into two equal right triangles.
According to the Pythagorean theorem, we have:
(the length of one side)^2 + (the length of the other side)^2 = (diagonal length)^2
In this case, both sides of the square are equal in length, so we can rewrite the equation as:
s^2 + s^2 = (diagonal length)^2
Simplifying this equation, we get:
2s^2 = (diagonal length)^2
Now, we can substitute the given diagonal length:
2s^2 = 8^2
2s^2 = 64
Dividing both sides by 2, we get:
s^2 = 32
Taking the square root of both sides, we find:
s ≈ √32
s ≈ 5.66 cm
Therefore, the length of each side of the square is approximately 5.66 cm.
To find the area of the square, we can square the length of one side:
Area = (side length)^2
Area = (5.66 cm)^2
Area ≈ 32.06 cm^2
Therefore, the area of the square with a diagonal of 8 cm is approximately 32.06 square centimeters.