To find the length of a chord that connects two perpendicular radii of a sphere, we can use some basic geometry.
1. **Understanding the Setup**: The two radii, being perpendicular, form a right triangle with the chord as the hypotenuse. Each radius measures 6 inches, and they meet at the center of the sphere.
2. **Labeling the Triangle**: Let’s label the center of the sphere as point O, the endpoints of the radii as point A and point B, and the points where the chord meets these radii as point C and point D. Therefore, AC and BD are both radii of the sphere. Since they are perpendicular, we can visualize this setup as a right triangle, OAC, where:
- OA = 6 inches (radius 1)
- OB = 6 inches (radius 2)
- OC is the distance from O to the chord (which we will find next).
3. **Finding the Length of OC**: The distance from the center O to the chord (OC) can be computed using the Pythagorean theorem in triangle OAC. Thus, for a right triangle, we have:
OC² + AC² = OA².
Since OA = 6 and AC (half the length of the chord, which we denote as x) is unknown, we rewrite it as:
OC² + x² = 6²
4. **Finding OC**: Since OC is the line connecting the center to the chord, and because the two radii are perpendicular, the correct geometric interpretation tells us that OC will also be a line from O perpendicular to the chord. This means OC can be found using the formula for the radius’s perpendicular distance given radii are equal. Here, OC can be computed as:
OC = 6 * cos(45°) = 6 * (√2/2) = 6√2/2 = 3√2 inches.
5. **Using OC to find Length of the Chord**: Now to find the length of the chord (CD), we can relate it to OC:
CD = 2 * √(OA² – OC²) where OA = 6:
CD = 2 * √(6² – (3√2)²).
6. **Calculating the Final Length of CD**: This simplifies to:
CD = 2 * √(36 – 18) = 2 * √18 = 2 * 3√2 = 6√2 inches.
Thus, the length of the chord connecting the two perpendicular radii is approximately 8.49 inches (if we convert 6√2 to decimal form), which provides a practical result in terms of measurement. Therefore, the answer to the question is:
Final Answer: 6√2 inches (approximately 8.49 inches).