एका वर्तुळाची त्रिज्या 8 सेमी आहे. एक जीवा AB अशी काढली आहे की... | Mathematics MCQ (मराठी) | Examoogle

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एका वर्तुळाची त्रिज्या 8 सेमी आहे. एक जीवा AB अशी काढली आहे की केंद्र O पासून जीवेपर्यंतचे अंतर 6 सेमी आहे. AB ची लांबी किती आहे?

$4\sqrt{15} \text{ सेमी}$

$4\sqrt{7} \text{ सेमी}$

$8\sqrt{7} \text{ सेमी}$

$2\sqrt{7} \text{ सेमी}$

Correct Answer

📐 MA • Math Direct FormulaMathematicsGeometry

$4\sqrt{15} \text{ सेमी}$

Quick Summary:

Use the Pythagorean triplet logic for right triangles: half-chord length is $\sqrt{8^2 - 6^2} = \sqrt{64 - 36} = \sqrt{28} = 2\sqrt{7}$ . The full chord is $2 \times 2\sqrt{7} = 4\sqrt{7}$ cm.

📐MAMath SolutionDirect Formula

📋 Given

Radius of the circle R = 8 cm, Distance from center to chord d = 6 cm.

🔢 Formula Used

L=2×R2−d2L = 2 \times \sqrt{R^2 - d^2}L=2×R2−d2​

⚡ Exam Hall Shortcut / Speed Trick

Use the Pythagorean triplet logic for right triangles: half-chord length is $\sqrt{8^2 - 6^2} = \sqrt{64 - 36} = \sqrt{28} = 2\sqrt{7}$ . The full chord is $2 \times 2\sqrt{7} = 4\sqrt{7}$ cm.

⚠️ Common Student Trap / Pitfall

Many students confuse the chord length with the half-chord length ( $\sqrt{28}$ ) and fail to multiply by 2.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

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+(L2)2r^2 = d^2 + (\frac{L}{2})^2r2=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=27x = \sqrt{8^2 - 6^2} = \sqrt{64 - 36} = \sqrt{28} = 2\sqrt{7}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=47AB = 2 \times 2\sqrt{7} = 4\sqrt{7}AB=2×27​=47​

✅

B is correct because the chord length is calculated as $2 \times \sqrt{8^2 - 6^2} = 4\sqrt{7}$ cm.

Core Concepts Used

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Circle Geometry Pythagorean Theorem Chord Properties

💡 EXAM TIP

This concept of right triangles within circles is frequently tested in trigonometry and coordinate geometry problems involving tangents and secants.

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