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Examoogle • User • info@examoogle.com • EE-2024-8821
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একটি বৃত্তের ব্যাসার্ধ 8 সেমি। একটি জ্যা AB এমনভাবে আঁকা হয়েছে যে কেন্দ্র O থেকে জ্যা-এর দূরত্ব 6 সেমি। AB-এর দৈর্ঘ্য কত?
415 সেমি
47 সেমি
87 সেমি
27 সেমি
415 সেমি
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.