To determine the distance between the center of a circle and a chord that is 30 cm long, follow these steps:
1. Understanding the Problem:
We need to find the perpendicular distance from the center of the circle to the chord. This distance is the shortest distance from the center to the chord.
2. Given Information:
- The length of the chord is 30 cm.
- We need to assume a radius for the circle since it was not provided. For practical calculations and to solve the problem, let's assume the radius of the circle is 17 cm.
3. Definition:
The perpendicular distance from the center of the circle to the chord can be calculated using the formula derived from the Pythagorean theorem applied to the right triangle formed by the radius (from the center to the edge where the chord meets the circle), half of the chord, and the perpendicular distance from the center to the chord.
[tex]\[
\text{distance} = \sqrt{\text{radius}^2 - (\text{half chord length})^2}
\][/tex]
4. Calculations:
- First, calculate half of the chord length:
[tex]\[
\text{half chord length} = \frac{\text{chord length}}{2} = \frac{30 \text{ cm}}{2} = 15 \text{ cm}
\][/tex]
- Use the formula to find the distance:
[tex]\[
\text{distance} = \sqrt{17^2 \text{ cm}^2 - 15^2 \text{ cm}^2}
\][/tex]
[tex]\[
\text{distance} = \sqrt{289 \text{ cm}^2 - 225 \text{ cm}^2}
\][/tex]
[tex]\[
\text{distance} = \sqrt{64 \text{ cm}^2}
\][/tex]
[tex]\[
\text{distance} = 8 \text{ cm}
\][/tex]
So, the distance between the center of the circle and the chord 30 cm long is 8 cm.
