Three circles with radii 3,4 and 5 touch each other externally. If P is the only point of intersection of tangents to these circles, find the distance of P from the points of contact.

Options

A. `sqrt5`
B. `sqrt6`
C. `sqrt7`
D. 3

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the Radii and Centers
We have three circles with radii:
- Circle 1: Radius \( r_1 = 3 \)
- Circle 2: Radius \( r_2 = 4 \)
- Circle 3: Radius \( r_3 = 5 \)

Let the centers of these circles be \( A \), \( B \), and \( C \) respectively.

### Step 2: Calculate the Lengths of the Sides of Triangle ABC
The distances between the centers of the circles (the sides of triangle \( ABC \)) can be calculated as follows:
- \( AB = r_1 + r_3 = 3 + 5 = 8 \)
- \( AC = r_2 + r_3 = 4 + 5 = 9 \)
- \( BC = r_1 + r_2 = 3 + 4 = 7 \)

Thus, the sides of triangle \( ABC \) are:
- \( AB = 8 \)
- \( AC = 9 \)
- \( BC = 7 \)

### Step 3: Calculate the Semi-Perimeter of Triangle ABC
The semi-perimeter \( s \) of triangle \( ABC \) is given by:
\[ s = \frac{AB + AC + BC}{2} = \frac{8 + 9 + 7}{2} = \frac{24}{2} = 12 \]

### Step 4: Calculate the Area of Triangle ABC Using Heron's Formula
Using Heron's formula, the area \( A \) of triangle \( ABC \) is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a = BC = 7 \), \( b = AC = 9 \), and \( c = AB = 8 \).

Substituting the values:
\[ A = \sqrt{12(12-7)(12-9)(12-8)} = \sqrt{12 \times 5 \times 3 \times 4} \]
Calculating the terms:
\[ = \sqrt{12 \times 60} = \sqrt{720} = \sqrt{144 \times 5} = 12\sqrt{5} \]

### Step 5: Calculate the Radius of the Incircle
The radius \( r \) of the incircle (the distance from point \( P \) to the points of contact) is given by the formula:
\[ r = \frac{A}{s} \]
Substituting the values we found:
\[ r = \frac{12\sqrt{5}}{12} = \sqrt{5} \]

### Conclusion
The distance of point \( P \) from the points of contact of the tangents to the circles is \( \sqrt{5} \).

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