Find the length of d in simplest radical form of he diagonal of a cube with sides of s units.
To find the length of the diagonal (d) of a cube with sides of length (s), we can use the Pythagorean theorem.
In a cube, the diagonal of a face forms a right triangle with two sides of the cube. The length of each side (s) is the same, and the diagonal (d) is the hypotenuse of this right triangle.
Let's consider one of the right triangles formed by the diagonal and two sides of the cube:
____d___
/ /
/ /
/ /
s /____/
Using the Pythagorean theorem, we have:
d² = s² + s²
Simplifying,
d² = 2s²
To find the length of d, we need to take the square root of both sides:
√(d²) = √(2s²)
d = √2s
Therefore, the length of d, in simplest radical form, is √2s units.
To find the length of the diagonal of a cube, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In the case of a cube, let's imagine a diagonal that passes through the center of the cube and connects two opposite corners. This diagonal forms a right triangle with two sides of equal length, which are the sides of the cube (s units), and the diagonal as the hypotenuse.
So, using the Pythagorean theorem, we have:
(diagonal)^2 = (side)^2 + (side)^2
Or
(d)^2 = s^2 + s^2
Simplifying the equation, we have:
d^2 = 2s^2
To find the length of the diagonal (d), we need to simplify the square root of 2s^2 to its simplest radical form.
Taking the square root of both sides, we get:
d = √(2s^2)
Since we want the answer in the simplest radical form, we cannot simplify the square root any further. Therefore, the length of the diagonal of the cube in its simplest radical form is √(2s^2).
diagonal of a square is √(s^2+s^2) = s√2
diagonal of a cube is √(s^2+s^2+s^2) = s√3
Think of it as the hypotenuse of a triangle perpendicular to the base, where one leg is the base diagonal (s√2) and the height is s, making the hypotenuse
h^2 = 2s^2 + s^2 = 3s^2