To find the length of the common external tangent between two circles with given radii and the distance between their centers, we can use a specific formula. Let's follow the steps to solve this problem.
### Given:
- Radius of the first circle, [tex]\( r_1 \)[/tex] = 5 cm
- Radius of the second circle, [tex]\( r_2 \)[/tex] = 12 cm
- Distance between the centers of the two circles, [tex]\( d \)[/tex] = 25 cm
### Formula:
The length of the common external tangent (L) can be found using the formula:
[tex]\[ L = \sqrt{d^2 - (r_1 + r_2)^2} \][/tex]
### Step-by-Step Solution:
1. Identify the variables:
- [tex]\( r_1 = 5 \)[/tex] cm
- [tex]\( r_2 = 12 \)[/tex] cm
- [tex]\( d = 25 \)[/tex] cm
2. Calculate the sum of the radii:
[tex]\[ r_1 + r_2 = 5 + 12 = 17 \text{ cm} \][/tex]
3. Square the distance between the centers:
[tex]\[ d^2 = 25^2 = 625 \][/tex]
4. Square the sum of the radii:
[tex]\[ (r_1 + r_2)^2 = 17^2 = 289 \][/tex]
5. Apply the formula to find the length of the common external tangent:
[tex]\[ L = \sqrt{625 - 289} \][/tex]
6. Subtract the squared values:
[tex]\[ 625 - 289 = 336 \][/tex]
7. Take the square root:
[tex]\[ \sqrt{336} \approx 18.33 \][/tex]
### Conclusion:
The length of the common external tangent to the two circles is approximately 18.33 cm. Since this length does not match any of the answer choices provided (a) 26 cm, (b) 24 cm, (c) 20 cm, (d) 22 cm, we do not have the correct rounding in the answer choices. However, the accurate calculated length is 18.33 cm.
