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A circle has a radius of 8 cm. A chord AB is drawn such that the distance from the centre O to the chord is 6 cm. What is the length of AB?
415 cm
47 cm
87 cm
27 cm
415 cm
Use the Pythagorean triplet logic for right triangles: half-chord length is 82−62=64−36=28=27. The full chord is 2×27=47 cm.
Radius of the circle R = 8 cm, Distance from center to chord d = 6 cm.
L=2×R2−d2
Use the Pythagorean triplet logic for right triangles: half-chord length is 82−62=64−36=28=27. The full chord is 2×27=47 cm.
Many students confuse the chord length with the half-chord length (28) and fail to multiply by 2.
Define Geometric Relationship
A perpendicular from the center of a circle to a chord bisects the chord. This forms a right-angled triangle with the radius as the hypotenuse, the distance from the center as one leg, and half the chord as the other.
r2=d2+(2L)2
Calculate Half-Chord Length
Substitute the given values R=8 and d=6 into the Pythagorean theorem to find the half-chord length (x).
x=82−62=64−36=28=27
Determine Full Chord Length
Multiply the half-chord length by 2 to get the total length of chord AB.
AB=2×27=47
B is correct because the chord length is calculated as 2×82−62=47 cm.
This concept of right triangles within circles is frequently tested in trigonometry and coordinate geometry problems involving tangents and secants.