At point A on the diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. Find the length of the chord CD parallel to XY at a distance of 16 cm from A.
Generated by claude-sonnet-4-6 · 2026-06-15 10:22 · grounding rag
Model Answer
Let centre O lie on diameter AB, with A on the circle (radius = 10 cm), so OA = 10 cm.
Tangent XAY is at A, perpendicular to OA (radius ⊥ tangent).
Chord CD is parallel to XY at distance 16 cm from A, so its distance from centre O = 16 − 10 = 6 cm.
Let M be the midpoint of CD. Then OM ⊥ CD and OM = 6 cm.
By Pythagoras in △OMC:
$$MC = \sqrt{OC^2 - OM^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ cm}$$
$$\therefore CD = 2 \times MC = 16 \text{ cm}$$
Source: Chapter 10, Circles
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Explanation
- The tangent at A is perpendicular to radius OA, so the 16 cm distance from A to chord CD means the perpendicular distance from centre O to CD is 16 − 10 = 6 cm.
- The perpendicular from centre bisects the chord (key property). Use Pythagoras with radius = 10 cm and half-distance = 6 cm to get half-chord = 8 cm, so full chord = 16 cm.
- Show the diagram setup clearly and the Pythagoras step for full marks.