In Fig., if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then C

1.	In Fig., if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to:(A) 2 cm(B) 3 cm(C) 4 cm(D) 5 cm
Answer» (A) 2 cm Explanation: Given: Radius of the circle = r = AO = 5 cm Length of chord AB = 8 cm Since the line drawn through the center of a circle to bisect a chord is perpendicular to the chord, therefore AOC is a right angled triangle with C as the bisector of AB. ∴ AC = ½ (AB) = 8/2 = 4 cm In right angled triangle AOC, by Pythagoras theorem, we have: (AO)² = (OC)² + (AC)² ⇒ (5)² = (OC)² + (4)² ⇒ (OC)² = (5)² – (4)² ⇒ (OC)² = 25 – 16 ⇒ (OC)² = 9 Take square root on both sides: ⇒ (OC) = 3 ∴ The distance of AC from the center of the circle is 3 cm. Now, OD is the radius of the circle, ∴ OD = 5 cm CD = OD – OC CD = 5 – 3 CD = 2 Therefore, CD = 2 cm Hence, option A is the correct answer.

In Fig., if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to:(A) 2 cm(B) 3 cm(C) 4 cm(D) 5 cm

Answer»

(A) 2 cm

Explanation:

Given:

Radius of the circle = r = AO = 5 cm

Length of chord AB = 8 cm

Since the line drawn through the center of a circle to bisect a chord is perpendicular to the chord, therefore AOC is a right angled triangle with C as the bisector of AB.

∴ AC = ½ (AB) = 8/2 = 4 cm

In right angled triangle AOC, by Pythagoras theorem, we have:

(AO)² = (OC)² + (AC)²

⇒ (5)² = (OC)² + (4)²

⇒ (OC)² = (5)² – (4)²

⇒ (OC)² = 25 – 16

⇒ (OC)² = 9

Take square root on both sides:

⇒ (OC) = 3

∴ The distance of AC from the center of the circle is 3 cm.

Now, OD is the radius of the circle, ∴ OD = 5 cm

CD = OD – OC

CD = 5 – 3

CD = 2

Therefore, CD = 2 cm

Hence, option A is the correct answer.

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