Two circles with centres A and B of radii 3 cm and 4 cm respectively intersect at two points C and D, such that AC and BC are tangents to the two circle. Find the length of the common chord CD.

Options

A. 4.2 cm
B. 8.4 cm
C. 2.4 cm
D. 4.8 cm

Answer

To find the length of the common chord CD of two intersecting circles with centers A and B, we can follow these steps:

### Step 1: Understand the Geometry
We have two circles with centers A and B. The radius of circle A is 3 cm and the radius of circle B is 4 cm. The points C and D are the intersection points of the two circles, and AC and BC are tangents to the respective circles.

### Step 2: Identify the Triangle
Since AC and BC are tangents to the circles, we can form triangle ABC where:
- AC = 3 cm (radius of circle A)
- BC = 4 cm (radius of circle B)

### Step 3: Use Pythagorean Theorem
In triangle ABC, we can find the length of AB using the Pythagorean theorem:
\[ AB = \sqrt{AC^2 + BC^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm} \]

### Step 4: Find the Area of Triangle ABC
We can calculate the area of triangle ABC using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Taking BC as the base and AC as the height:
\[ \text{Area} = \frac{1}{2} \times BC \times AC = \frac{1}{2} \times 4 \times 3 = 6 \text{ cm}^2 \]

### Step 5: Relate Area to CE
The area can also be expressed using the height from point C to line AB. Let CE be the height from point C to line AB. Then, the area can also be written as:
\[ \text{Area} = \frac{1}{2} \times AB \times CE = \frac{1}{2} \times 5 \times CE \]
Setting the two area expressions equal to each other:
\[ 6 = \frac{1}{2} \times 5 \times CE \]

### Step 6: Solve for CE
Multiplying both sides by 2:
\[ 12 = 5 \times CE \]
Dividing by 5:
\[ CE = \frac{12}{5} = 2.4 \text{ cm} \]

### Step 7: Find the Length of the Common Chord CD
Since E is the midpoint of the common chord CD, the length of CD is twice CE:
\[ CD = 2 \times CE = 2 \times 2.4 = 4.8 \text{ cm} \]

### Final Answer
The length of the common chord CD is **4.8 cm**.

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