Question

The radii of two concentric circles are 15 cm and 20 cm. A line segment ABCD cuts the outer circle at A and D and inner circle at B and C. If BC = 18 cm, find the length of AB.

Answer 1

Given:

• Radius of inner circle = 15 cm
• Radius of outer circle = 20 cm
• BC = 18 cm
• O = Common center of both circles

Step 1: Find the distance of the chord from the center

For the inner circle:

`BC = 2sqrt(15^2 - x^2)`

Substitute BC = 18:

`18 = 2sqrt(225 - x^2)`

`9 = sqrt(225 - x^2)`

225 – x² = 81

x² = 144

⇒ x = 12 m

Step 2: Find the length of the outer chord AD

For the outer circle:

`AD = 2sqrt(20^2 - x^2)`

`AD = 2sqrt(400 - 144)`

= `2sqrt(256)`

= 2 × 16

= 32 cm

Step 3: Find AB

The line segment AD is divided into:

• AB (outer part before entering inner circle)
• BC (inside inner circle)
• CD (outer part after leaving inner circle)

Since the circles are concentric, AB = CD.

Thus:

`AB = (AD - BC)/2`

`AB = (32 - 18)/2`

= `14/2`

= 7 cm