Question

AB and CD are two chords of a circle such that AB = 6 cm, CD = 12 cm and AB||CD. If the distance between AB and CD is 3 cm, find the radius​

Answer 1

Step-by-step explanation:

Solution:-

Let radius be r cm

PQ = 3 cm

Also,

Let OQ be x cm

Then, OP = x + 3 cm

AP = ½ AB = 3 cm

Also,

CQ = ½ CD = 6 cm

Applying Pythagoras theorem in right triangles OAP and OCQ, we obtain

OA² = OP² + AP² and OC² = OQ² + CQ²

→ r² = (x + 3)² + 3² and r² = x² + 6²

→ (x + 3)² + 3² = x² + 6²

→ x² + 6x + 9 + 9 = x² + 36

→ 6x = 18

→ x = 3

Putting the value of x in r² = x² + 6², we get

r² = 3² + 6² = 45

r = 6.7 cm (approximately)

$\red{ \frak{Hence \: answered}}$

Answer 2

Given :-

• AB and CD are two chords of circle such that AB = 6 and CD 12 and AB || CD . The difference between its chords of circle is 3 cm .

To Find :-

• The radius of this circle = ?

Solution and concept :-

To calculate the radius of circle at first we have to notice in the given figure. By noticing we get that PQ = 3cm which is the distance between the chord of Circle AB and CD. OP is perpendicular on CD and AB such OP becomes the collinear line that intersect at point Q and P. As we know that Perpendicular bisect the Sides into equal parts. AB= AP + PB and CD = CQ + QD and AP = PB(OP bisect AB) and CQ = QD (OP bisect CD) . Here we have given radius of circle is r. By Applying Pythagoras theorem to calculate the radius of circle by comparing equation.

Calculation begins :-

• Let's focus on ∆ OAP and ∆ OCQ :-

⇒ In right angle triangle OAP :-

• OP = OQ + 3
• OA = Radius ( r)
• AP = 1/2 × AB = 3cm

By using Pythagoras theorem :-

⇒ OA² = OP² + AP²

⇒ r² = (OQ + 3)² + 3²

⇒ r² = OQ² + 6OQ + 9 + 9

⇒ r² = OQ² + 6OQ + 18 ---------(i)

⇒ In right angle triangle OCQ :-

• OQ = OQ
• OC = Radius (r)
• CQ = 1/2 × CD = 6 cm

By using Pythagoras theorem :-

⇒OC² = CQ² + OQ²

⇒ r² = 6² + OQ² -------(ii)

• Substituting the value of r² in (ii)

⇒ 36 + OQ² = OQ² + 6OQ + 18

⇒ 6OQ = 36 - 18

⇒ 6OQ = 18

⇒ OQ = 3

Now calculate the radius of circle :-

• From eq (ii) putting the value of OQ :-

⇒ r² = 6² + OQ²

⇒ r² = 36 + 3²

⇒ r² = 36 + 9

⇒ r² = 45.

⇒ r = √45 ⇒r = 6.70 cm

Hence,

• The radius of circle (r) = 6.70 cm