Key Concepts and Formulas

• Orthogonal Circles: Two circles intersect orthogonally if the tangents at their point of intersection are perpendicular. This implies that the radii drawn to the point of intersection are also perpendicular.
• Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$).
• Area of a Triangle: Area can be calculated as $\frac{1}{2} \times \text{base} \times \text{height}$.

Step-by-Step Solution

Step 1: Understand the Geometry of Orthogonal Intersection

• We are given two circles, $C_1$ and $C_2$, with radii $r_1 = 5$ cm and $r_2 = 12$ cm, respectively, and centers $O_1$ and $O_2$. They intersect at points $A$ and $B$. The angle of intersection is $90^\circ$, meaning the circles intersect orthogonally.
• Why this is important: Orthogonal intersection means the radii at the point of intersection are perpendicular. Therefore, $\angle O_1 A O_2 = 90^\circ$, and $\triangle O_1 A O_2$ is a right-angled triangle.
• Thus, $\triangle O_1 A O_2$ is a right-angled triangle with $O_1A = r_1 = 5$ cm and $O_2A = r_2 = 12$ cm.

Step 2: Calculate the Distance Between the Centers ($O_1O_2$)

• In the right-angled triangle $\triangle O_1 A O_2$, we need to find the length of the hypotenuse $O_1O_2$.
• Why Pythagorean theorem? Since $\triangle O_1 A O_2$ is a right-angled triangle, we can apply the Pythagorean theorem to find the distance between the centers.
• Applying the Pythagorean theorem: $(O_1O_2)^2 = (O_1A)^2 + (O_2A)^2$ $(O_1O_2)^2 = r_1^2 + r_2^2$ $(O_1O_2)^2 = 5^2 + 12^2$ $(O_1O_2)^2 = 25 + 144$ $(O_1O_2)^2 = 169$ $O_1O_2 = \sqrt{169}$ $O_1O_2 = 13 \text{ cm}$ The distance between the centers of the two circles is 13 cm.

Step 3: Relate the Common Chord to the Right-Angled Triangle and Find AM

• Let $M$ be the midpoint of the common chord $AB$. The line segment $O_1O_2$ bisects the common chord $AB$ at a right angle. Thus, $AM$ is the altitude from vertex $A$ to the hypotenuse $O_1O_2$ in $\triangle O_1AO_2$.
• Why use area? The area of $\triangle O_1AO_2$ can be calculated in two ways: using the legs $O_1A$ and $O_2A$, and using the hypotenuse $O_1O_2$ and the altitude $AM$. Equating these gives us a way to find $AM$.
• Area of $\triangle O_1AO_2$ using legs $O_1A$ and $O_2A$: $\text{Area} = \frac{1}{2} \times O_1A \times O_2A = \frac{1}{2} r_1 r_2$ $\text{Area} = \frac{1}{2} \times 5 \times 12 = 30 \text{ cm}^2$
• Area of $\triangle O_1AO_2$ using hypotenuse $O_1O_2$ and altitude $AM$: $\text{Area} = \frac{1}{2} \times O_1O_2 \times AM$
• Equating the two expressions for the area: $\frac{1}{2} r_1 r_2 = \frac{1}{2} O_1O_2 \times AM$ $r_1 r_2 = O_1O_2 \times AM$

Step 4: Calculate the Length of the Common Chord ($AB$)

• Substitute the known values into the equation from Step 3:
  • $r_1 = 5$ cm
  • $r_2 = 12$ cm
  • $O_1O_2 = 13$ cm (from Step 2) $5 \times 12 = 13 \times AM$ $60 = 13 \times AM$ $AM = \frac{60}{13} \text{ cm}$
• Why multiply by 2? Remember that $AM$ is only half the length of the common chord $AB$.
• Therefore, the length of the common chord $AB$ is: $AB = 2 \times AM$ $AB = 2 \times \frac{60}{13}$ $AB = \frac{120}{13} \text{ cm}$

Common Mistakes & Tips

• Diagram: Always draw a diagram to visualize the problem. This helps in understanding the relationships between the radii, the distance between the centers, and the common chord.
• Orthogonality: Remember that orthogonal intersection implies perpendicular radii at the point of intersection.
• Altitude Formula: Recall that in a right triangle, the area can be expressed as half the product of the legs or half the product of the hypotenuse and the altitude to the hypotenuse.

Summary

The problem involves two circles intersecting orthogonally. This means the radii at the point of intersection are perpendicular, forming a right-angled triangle. By using the Pythagorean theorem, we found the distance between the centers. Then, by equating two expressions for the area of the triangle, we found half the length of the common chord. Doubling this gave us the length of the common chord.

The final answer is $\boxed{\frac{120}{13}}$, which corresponds to option (C).