The chord of a circle of radius 17 cm is 30 cm long. Calculate the distance of the chord from the centre of the circle.

Answer Details

To find the distance from the center of the circle to the chord, visualize or draw the circle. Let’s call the center of the circle O, the chord AB, and let OM be the perpendicular from O to AB, where M is the midpoint of AB.

Key concepts:

• The perpendicular from the center of a circle to a chord bisects the chord, meaning it cuts the chord exactly in half.
• We can use the right triangle OMA (O is the center, M is the midpoint of the chord, A is at one end of the chord) and the Pythagorean Theorem.

Given data:

• Radius of the circle, OA = 17 cm (since OA is a radius)
• Length of chord AB = 30 cm

Step-by-step solution:

• Since M is the midpoint of AB, AM = AB / 2 = 30/2 = 15 cm.
• The triangle OMA is a right triangle, where:
  • OA = 17 cm (radius, hypotenuse)
  • AM = 15 cm (half of the chord, one leg)
  • OM = ? (distance from center to the chord, the other leg)
• By the Pythagorean Theorem: \[ OA^2 = OM^2 + AM^2 \] \[ 17^2 = OM^2 + 15^2 \] \[ 289 = OM^2 + 225 \] \[ OM^2 = 289 - 225 = 64 \] \[ OM = \sqrt{64} = 8~\text{cm} \]

Therefore, the distance from the center of the circle to the chord is 8 cm.
This is because, in a circle, drawing a perpendicular from the center to the chord splits the chord into two equal segments and creates a right triangle, allowing the use of the Pythagorean Theorem to solve for the unknown distance.