Two circles of radii 5 cm and 3 cm intersect at two distinct points. Their centres are 4 cm apart. Find the length of their common chord.

Answer

To find the length of the common chord of two intersecting circles with given radii and distance between their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Radius of the first circle (R1) = 5 cm
- Radius of the second circle (R2) = 3 cm
- Distance between the centers of the circles (d) = 4 cm

2. **Draw the Diagram:**
- Draw two circles with centers P and Q.
- Mark the points of intersection as A and B.
- Draw the line segment PQ (the distance between the centers) and the common chord AB.

3. **Label the Points:**
- Let R be the midpoint of the chord AB.
- Since PR is perpendicular to AB, we can denote the lengths:
- PR = x cm (the distance from center P to the midpoint R)
- AR = RB = y cm (the distance from R to points A and B)

4. **Apply the Pythagorean Theorem:**
- In triangle PAR (where P is the center of the first circle):
\[ PA^2 = PR^2 + AR^2 \]
Substituting the known values:
\[ 5^2 = x^2 + y^2 \quad \text{(1)} \]
This simplifies to:
\[ 25 = x^2 + y^2 \]

- In triangle QAR (where Q is the center of the second circle):
\[ QA^2 = QR^2 + AR^2 \]
Here, QR = d - PR = 4 - x:
\[ 3^2 = (4 - x)^2 + y^2 \quad \text{(2)} \]
This simplifies to:
\[ 9 = (4 - x)^2 + y^2 \]

5. **Expand and Simplify Equation (2):**
\[ 9 = (16 - 8x + x^2) + y^2 \]
Rearranging gives:
\[ 9 = 16 - 8x + x^2 + y^2 \]
\[ 8x = 16 + y^2 - 9 \]
\[ 8x = 7 + y^2 \quad \text{(3)} \]

6. **Substitute Equation (1) into Equation (3):**
From equation (1), we have \( y^2 = 25 - x^2 \).
Substitute this into equation (3):
\[ 8x = 7 + (25 - x^2) \]
\[ 8x = 32 - x^2 \]
Rearranging gives:
\[ x^2 + 8x - 32 = 0 \]

7. **Solve the Quadratic Equation:**
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-32)}}{2 \cdot 1} \]
\[ x = \frac{-8 \pm \sqrt{64 + 128}}{2} \]
\[ x = \frac{-8 \pm \sqrt{192}}{2} \]
\[ x = \frac{-8 \pm 8\sqrt{3}}{2} \]
\[ x = -4 \pm 4\sqrt{3} \]
Since distance cannot be negative, we take:
\[ x = -4 + 4\sqrt{3} \]

8. **Find y using Equation (1):**
Substitute x back into equation (1):
\[ y^2 = 25 - (-4 + 4\sqrt{3})^2 \]
Calculate \( (-4 + 4\sqrt{3})^2 \):
\[ = 16 - 32\sqrt{3} + 48 = 64 - 32\sqrt{3} \]
Thus,
\[ y^2 = 25 - (64 - 32\sqrt{3}) = 25 - 64 + 32\sqrt{3} = -39 + 32\sqrt{3} \]

9. **Calculate the Length of the Common Chord AB:**
The length of the common chord AB is:
\[ AB = 2y = 2\sqrt{y^2} \]
Substitute the value of y:
\[ = 2\sqrt{-39 + 32\sqrt{3}} \]

### Final Result:
The length of the common chord AB is \( 2\sqrt{-39 + 32\sqrt{3}} \) cm.