AB is the chord of the circle of centre O and the radius is 17 cm. If OM ⊥ AB and OM = 8 cm then the length of the chord AB is

• (a) 12 cm
• (b) 30 cm
• (c) 15 cm
• (d) 24 cm

To find the length of the chord AB in the circle with center O and radius 17 cm, we can use the relationship between the radius, the perpendicular distance from the center to the chord, and the chord itself.

Given Information

• Radius (r) = 17 cm
• Distance from center to chord (OM) = 8 cm

Using the Right Triangle

When you draw a line from the center O to the midpoint M of the chord AB, you create a right triangle OMA, where:

• OM is the height (8 cm)
• OA is the radius (17 cm)
• AM is half the length of the chord AB

Applying the Pythagorean Theorem

In triangle OMA, we can apply the Pythagorean theorem:

OA² = OM² + AM²

Substituting the known values:

17² = 8² + AM²

289 = 64 + AM²

AM² = 289 - 64

AM² = 225

AM = √225 = 15 cm

Finding the Length of Chord AB

Since AM is half of AB, the full length of the chord AB is:

AB = 2 × AM = 2 × 15 cm = 30 cm

Therefore, the length of the chord AB is 30 cm, which corresponds to option (b).