એક વર્તુળની ત્રિજ્યા 8 સેમી છે. એક જીવા AB એવી રીતે દોરવામાં આ... | Mathematics MCQ (ગુજરાતી) | Examoogle

MathematicsGeometry

એક વર્તુળની ત્રિજ્યા 8 સેમી છે. એક જીવા AB એવી રીતે દોરવામાં આવે છે કે કેન્દ્ર O થી જીવાનું અંતર 6 સેમી છે. AB ની લંબાઈ કેટલી છે?

$4\sqrt{15} \text{ cm}$

$4\sqrt{7} \text{ cm}$

$8\sqrt{7} \text{ cm}$

$2\sqrt{7} \text{ cm}$

Correct Answer

📐 MA • Math Direct FormulaMathematicsGeometry

$4\sqrt{15} \text{ cm}$

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

Click any tag to open in AI Tutor

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.

Discussion (0)

Loading discussion...

PrevNext

Correct Answer

Related Questions

Discussion (0)

Correct Answer

Related Questions

Discussion (0)