To find the length of a chord connecting two perpendicular radii of a sphere, we can use some basic geometry. In this case, we have a sphere with a radius of 6 inches.
First, we recognize that two perpendicular radii create a right triangle when we draw the chord between their endpoints. The radius of the sphere (6 inches) serves as the hypotenuse of this triangle.
Since the two radii are perpendicular to each other, we can visualize one radius extending vertically and the other horizontally. The length of the chord we’re trying to find will form the base of this right triangle.
Using the Pythagorean theorem, we know that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case:
c = 6 inches (the radius)
a = b = x (the lengths of the two sides of the triangle formed by the radii)
According to the Pythagorean theorem:
c² = a² + b²
6² = x² + x²
36 = 2x²
x² = 18
x = √18 = 3√2
This means that each radius (side of the triangle) has a length of 3√2 inches. To find the length of the chord (which is the distance between the two points where the radii intersect the sphere), we can express this chord as:
Length of the chord = 2 * x
So, plugging in the value of x:
Length of the chord = 2 * 3√2 = 6√2 inches.
Thus, the length of the chord connecting two perpendicular radii of the sphere with a radius of 6 inches is 6√2 inches.