Question

Two concentric circles are of radii 5 cm and 4 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Hint:

Right triangles in concentric circle problems almost always involve standard Pythagorean triplets like (3, 4, 5).
Once you identify the half-chord is 3 cm, you can immediately double it to get the total chord length of 6 cm.

Answer 1

Step 1: Understanding the Question:
Concentric circles are circles that share the exact same centre but have different radii.
We have a larger circle of radius $5 \text{ cm}$ and a smaller circle of radius $4 \text{ cm}$.
We need to find the length of a chord of the larger circle that is tangent to (just touches) the smaller circle.

Step 2: Key Formula or Approach:
1. Let O be the common centre of the two circles.
2. Let AB be the chord of the larger circle that touches the smaller circle at point P.
3. Since AB touches the smaller circle, OP is the radius of the smaller circle and is perpendicular to the tangent chord AB ($OP \perp AB$).
4. Connect OA, which is the radius of the larger circle.
5. Use the Pythagorean theorem in right-angled triangle $\Delta\text{OPA}$ to find the length of AP.
6. The perpendicular from the centre of a circle to a chord bisects the chord, so $AB = 2 \times AP$.

Step 3: Detailed Explanation:

• Set up the geometric properties:
- Radius of the outer circle, $OA = 5 \text{ cm}$.
- Radius of the inner circle, $OP = 4 \text{ cm}$.
- The tangent-radius property guarantees that $\angle\text{OPA} = 90^\circ$.

• Apply the Pythagorean theorem to right-angled triangle $\Delta\text{OPA}$:
\[ OA^2 = OP^2 + AP^2 \]
\[ 5^2 = 4^2 + AP^2 \]
\[ 25 = 16 + AP^2 \]

• Solve for AP:
\[ AP^2 = 25 - 16 = 9 \]
\[ AP = \sqrt{9} = 3 \text{ cm} \]

• Use the chord bisector property:
The line segment OP is perpendicular to the chord AB.
Since the perpendicular from the centre to a chord bisects the chord:
\[ AP = PB \]
Therefore, the full length of the chord AB is:
\[ AB = 2 \times AP \]
\[ AB = 2 \times 3 = 6 \text{ cm} \]


Step 4: Final Answer:
The length of the chord of the larger circle is 6 cm.