Question 272 marksVery Short AnswerSection B

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

Q: 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.

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 $AB = 2 \times PA = 6\text{ cm}$.

Circles

Tangent to a Circle (Concentric Circles)

Official answer

OP \perp AB

concentric circles

tangent

chord

perpendicular bisector

Pythagoras theorem

3-4-5 triplet

radius

Marking scheme

1. $5^2 = 4^2 + PA^2$
2. $PA = 3\text{ cm}$

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

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

Common mistakes

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

Analysis and 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.

Interesting facts

$3^2 + 4^2 = 5^2$

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

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

Question details

Q27 of 56 questions

Board: CBSE
Class: Class 10
Subject: Mathematics (Standard)
Year: 2026
Section: Section B
Marks: 2
Type: Very Short Answer
Chapter: Circles
Topic: $Tangent to a Circle (Concentric Circles)$
Bloom's level: apply
Syllabus topic: $Unit IV: Geometry — Chapter: Circles (Tangent from an external point; chord of larger circle tangent to smaller concentric circle)$
Author: CBSE
Reviewed by: Padhantu Academic Team

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Frequently asked questions

$Why does the point of tangency bisect the chord?$

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

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

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