Question

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.

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

For two circles touching externally, use the formula $L = \sqrt{d^2 - (r_1 - r_2)^2}$ to find the length of the direct common tangent, where $d$ is distance between centers.

Answer 1

The Correct Option is A

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.