Question

Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is 

• A. \(\sqrt{2}\)
• B. \(\frac{\pi}{3}\)
• C. \(\frac{1}{\sqrt{2}}\)
• D. 1

Answer 1

The Correct Option is D

From the figure

SO=4-r.
 

 We are given a right-angled triangle POS. Using the Pythagorean theorem:

\[ (4 + r)^2 = 4^2 + (4 - r)^2 \]

Step 1: Expand both sides

Left-hand side: \[ (4 + r)^2 = 16 + 8r + r^2 \]

Right-hand side: \[ 4^2 + (4 - r)^2 = 16 + (16 - 8r + r^2) = 32 - 8r + r^2 \]

Step 2: Set both sides equal

\[ 16 + 8r + r^2 = 32 - 8r + r^2 \]

Step 3: Cancel and simplify

Subtract \( r^2 \) from both sides: \[ 16 + 8r = 32 - 8r \]

Bring terms together: \[ 8r + 8r = 32 - 16 \Rightarrow 16r = 16 \Rightarrow r = 1 \]

Final Answer:

\[ \boxed{r = 1} \]