Answer :
To determine the distance from the center of a circle with a given diameter to a chord of a specified length, we can use geometric properties of the circle and the right triangle formed by the radius, half the chord, and the perpendicular distance from the center to the chord.
Here's a step-by-step solution:
1. Find the radius of the circle:
The radius [tex]\( r \)[/tex] is half of the diameter.
[tex]\[ r = \frac{25}{2} = 12.5 \text{ cm} \][/tex]
2. Define the chord and its properties:
The length of the chord is given as 16 cm. Half of the chord length would be:
[tex]\[ \frac{16}{2} = 8 \text{ cm} \][/tex]
3. Use the Pythagorean theorem:
Consider the right triangle formed by:
- The radius of the circle [tex]\( r \)[/tex]
- Half the chord length [tex]\( \frac{16}{2} = 8 \text{ cm} \)[/tex]
- The perpendicular distance from the center to the chord, which we denote as [tex]\( d \)[/tex]
By the Pythagorean theorem:
[tex]\[ r^2 = d^2 + \left(\frac{\text{chord length}}{2}\right)^2 \][/tex]
Plugging in the values:
[tex]\[ 12.5^2 = d^2 + 8^2 \][/tex]
Calculate squares:
[tex]\[ 156.25 = d^2 + 64 \][/tex]
4. Solve for [tex]\( d^2 \)[/tex]:
[tex]\[ d^2 = 156.25 - 64 \][/tex]
[tex]\[ d^2 = 92.25 \][/tex]
5. Calculate [tex]\( d \)[/tex]:
[tex]\[ d = \sqrt{92.25} \][/tex]
[tex]\[ d \approx 9.6 \, \text{cm} \][/tex]
So, the distance from the center of the circle to the chord is approximately 9.6 cm. This distance is obtained using the geometric properties of circles and right triangles.
Here's a step-by-step solution:
1. Find the radius of the circle:
The radius [tex]\( r \)[/tex] is half of the diameter.
[tex]\[ r = \frac{25}{2} = 12.5 \text{ cm} \][/tex]
2. Define the chord and its properties:
The length of the chord is given as 16 cm. Half of the chord length would be:
[tex]\[ \frac{16}{2} = 8 \text{ cm} \][/tex]
3. Use the Pythagorean theorem:
Consider the right triangle formed by:
- The radius of the circle [tex]\( r \)[/tex]
- Half the chord length [tex]\( \frac{16}{2} = 8 \text{ cm} \)[/tex]
- The perpendicular distance from the center to the chord, which we denote as [tex]\( d \)[/tex]
By the Pythagorean theorem:
[tex]\[ r^2 = d^2 + \left(\frac{\text{chord length}}{2}\right)^2 \][/tex]
Plugging in the values:
[tex]\[ 12.5^2 = d^2 + 8^2 \][/tex]
Calculate squares:
[tex]\[ 156.25 = d^2 + 64 \][/tex]
4. Solve for [tex]\( d^2 \)[/tex]:
[tex]\[ d^2 = 156.25 - 64 \][/tex]
[tex]\[ d^2 = 92.25 \][/tex]
5. Calculate [tex]\( d \)[/tex]:
[tex]\[ d = \sqrt{92.25} \][/tex]
[tex]\[ d \approx 9.6 \, \text{cm} \][/tex]
So, the distance from the center of the circle to the chord is approximately 9.6 cm. This distance is obtained using the geometric properties of circles and right triangles.
