How many tangents can be drawn to two circles, whose radii are equal to 4 cm and 5 cm respectively and the distance between the two centres is 5 cm.

Options

A. 0
B. 1
C. 2
D. 3

Answer

To determine how many tangents can be drawn to two circles with given radii and the distance between their centers, we can follow these steps:

### Step 1: Identify the given values
- Radius of the first circle (r1) = 4 cm
- Radius of the second circle (r2) = 5 cm
- Distance between the centers of the two circles (d) = 5 cm

### Step 2: Analyze the relationship between the circles
To find the number of common tangents between two circles, we can use the following conditions:
1. If the circles are separate (d > r1 + r2), there are 4 tangents (2 external and 2 internal).
2. If the circles are touching externally (d = r1 + r2), there are 3 tangents (2 external and 1 internal).
3. If the circles are overlapping (|r1 - r2| < d < r1 + r2), there are 2 external tangents.
4. If the circles are touching internally (d = |r1 - r2|), there is 1 tangent.
5. If one circle is inside the other without touching (d < |r1 - r2|), there are no tangents.

### Step 3: Calculate the sum of the radii
Calculate r1 + r2:
- r1 + r2 = 4 cm + 5 cm = 9 cm

### Step 4: Compare the distance with the sum of the radii
Now, we compare the distance between the centers (d) with the sum of the radii:
- d = 5 cm
- r1 + r2 = 9 cm

Since d (5 cm) is less than r1 + r2 (9 cm), we check the next condition.

### Step 5: Check the difference of the radii
Calculate |r1 - r2|:
- |r1 - r2| = |4 cm - 5 cm| = 1 cm

Now, we check if the distance is greater than the absolute difference of the radii:
- |r1 - r2| < d < r1 + r2
- 1 cm < 5 cm < 9 cm

This condition holds true, which means there are 2 external tangents.

### Conclusion
Thus, the number of tangents that can be drawn to the two circles is **2**.

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