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.

Circles

Tangent to a Circle (Concentric Circles)

Official Answer

Length of chord = 6 cm. Since chord AB of the larger circle is tangent to the smaller circle at P, $OP \perp AB$ , and as OP passes through the common centre O, P bisects AB. In right triangle OPA, $OA^2 = OP^2 + PA^2 \implies 25 = 16 + PA^2 \implies PA = 3\text{ cm}$ . So $AB = 2 \times PA = 6\text{ cm}$ .

concentric circlestangentchordperpendicular bisectorPythagoras theorem3-4-5 tripletradius

Marking Scheme

11 mark: correct figure/reasoning — radius ⊥ tangent at point of contact, and this radius bisects the chord (half chord = PA); setting up Pythagoras theorem $5^2 = 4^2 + PA^2$ .
21 mark: correct computation $PA = 3\text{ cm}$ and doubling to get chord $AB = 6\text{ cm}$ as final answer.

Hint

Draw the two concentric circles; the tangent point on the smaller circle bisects the chord of the larger circle — use Pythagoras theorem in the right triangle formed by the two radii and half the chord.

Quick Oral Answer

The radius to the point where the chord touches the smaller circle is perpendicular to the chord and bisects it, so using Pythagoras theorem with radii 5 and 4, half the chord is 3 cm, making the full chord 6 cm.

Analysis & Explanation

Combine the tangent-perpendicularity theorem with the perpendicular-bisects-chord theorem to reduce this to a Pythagoras calculation.

Concept

The radius to the point of tangency is perpendicular to the tangent (chord).
A perpendicular from the centre to a chord bisects the chord.

Key Points

This gives a right triangle: hypotenuse = 5 (larger radius), one leg = 4 (smaller radius), other leg = half-chord = 3 (3-4-5 triple), so full chord = 6 cm.

Common Mistakes

Forgetting to double the half-chord PA, wrongly giving 3 cm as the final answer instead of 6 cm.

Real-world Application

This concentric-circle geometry is used in engineering, e.g., designing circular gears or pipe fittings.

Common Mistakes

1Forgetting to double the half-chord length PA, giving the final answer as 3 cm instead of 6 cm.
2Not recognizing that the radius to the point of tangency is perpendicular to the chord, and instead trying to use an incorrect or unrelated formula.
3Mixing up which radius (5 cm or 4 cm) is the hypotenuse versus the leg of the right triangle in the Pythagoras theorem step.

Interesting Facts

5, 4, 3 is the smallest and most famous Pythagorean triple ( $3^2 + 4^2 = 5^2$ ), which is why this exact combination of radii (5 cm and 4 cm) is a favourite in CBSE circle problems.

Concentric circle geometry (same principle used here) is applied in real engineering contexts such as designing washers, O-rings, and annular gears where an inner and outer circular boundary share a common centre.

The theorem 'perpendicular from centre bisects the chord' was known to Euclid and appears in Book III of his 'Elements' (circa 300 BCE), one of the oldest surviving systematic geometry texts.

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Frequently Asked Questions

Why does the point of tangency bisect the chord?

Because both circles share the same centre O, the line OP (radius of smaller circle to the tangency point) is perpendicular to the chord AB of the larger circle, and a perpendicular from the centre to any chord always bisects that chord.

Would the method change if the radii were not a Pythagorean triple?

No, the method (Pythagoras theorem in the right triangle formed) remains the same; only the final numeric value would involve a surd (square root) instead of a whole number.