Question

5 সেমি, দৈর্ঘ্যের ব্যাসার্ধের একটি বৃত্তে AB ও AC দুটি সমান দৈর্ঘ্যের জ্যা। বৃত্তের কেন্দ্র
ABC ত্রিভুজের বাইরে অবস্থিত। AB=AC=6 সেমি হলে, BC জ্যার দৈর্ঘ্য নির্ণয় করাে।​

Answer 1

Given:

AB and AC are two equal chords of a circle having a radius of 5 cm

The  centre of the circle is located outside the Δ ABC

AB = AC = 6 cm

To find:

The length of the chord BC

Solution:

From the below-attached figure, we can say,

• O is the centre of the circle.
• OB = radius of the circle = 5 cm  

• ∆ ABC is inscribed inside the circle such that AB = AC = 6 cm

Step 1:

We know that → the angle bisector of an angle between two equal chords of a circle passes through the centre of the circle.

Here, AB and AC are given as two equal chords of a circle, therefore, the centre of the circle O lies on the bisector of ∠BAC.  

⇒ OA is the bisector of ∠BAC

Let’s join the points B & C intersecting OA at P.

Again we know that → the internal bisector of an angle divides the opposite sides in the ratio of the sides containing the angle

⇒ bisector OA will divide BC in the ratio of AB : AC.

∴ The ratio in which P divides BC = 6 : 6 = 1 : 1

⇒ P is mid-point of BC

∴ CP = BP …… (i)

Since we know that → if a line from the centre to chord, divides the chord into two equal parts, then the line joining the chord will be perpendicular to it.

⇒ OP ⊥ BC  

Step 2:

Now,    

In right-angled triangle Δ ABP, by using the Pythagoras theorem, we get  

AB² = AP² + BP²  

⇒ BP² = AB² – AP²  

⇒ BP² = 6² - AP² ............. (ii)

And,

In right-angled triangle OBP, by using the Pythagoras theorem, we get  

OB² = OP² + BP²  

⇒ OB² = (AO - AP)² + BP²  

⇒ 5² = (5 - AP)² + BP²  

⇒ BP² = 25 - (5 - AP)² ........... (iii)

From equation (ii) and (iii), we get  

62 - AP² = 25 - (5 - AP)²  

⇒ 36 – AP² = 25 – (25 – 10AP + AP²)  

⇒ 11 – AP² = - 25 + 10AP – AP²

⇒ 36 = 10AP  

⇒ AP = 3.6 cm

Substituting the value of AP in (ii), we get  

BP² = 6² - (3.6)² = 23.04  

⇒ BP = 4.8 cm ….. (iv)

Therefore, from (i) and (iv), we get  

BC = 2 * BP = 2 * 4.8 = 9.6 cm

Thus, $\boxed{\bold{The\: length\: of\:the\:chord\: BC\: is\:\underline{ 9.6 \:cm}}}.$

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