Two circles with centers $O_1$ and $O_2$ touch externally. The radius of the first circle is 4 cm and the second is 9 cm. The distance between their centers is 13 cm. Find the length of the direct common tangent.
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- 0
- 1
- 2
- 3
The Correct Option is A
Solution and Explanation
To find the length of the direct common tangent between two externally touching circles, we can use the formula for the length of a tangent between two circles with radii \( r_1 \) and \( r_2 \) and distance \( d \) between their centers:
\( L = \sqrt{d^2 - (r_1 + r_2)^2} \).
Here, the given values are:
- Radius of the first circle, \( r_1 = 4 \, \text{cm} \)
- Radius of the second circle, \( r_2 = 9 \, \text{cm} \)
- Distance between the centers, \( d = 13 \, \text{cm} \)
Since the circles touch externally, the distance between their centers \( d = r_1 + r_2 = 4 + 9 = 13 \, \text{cm} \). Substituting these values into the tangent length formula gives:
\[ L = \sqrt{13^2 - (4 + 9)^2} \]
\[ L = \sqrt{169 - 169} \]
\[ L = \sqrt{0} \]
\[ L = 0 \]
Thus, the length of the direct common tangent is \( 0 \) cm.
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