Equal Chords Equidistant from Center (Motivate) MCQs Quiz | Class 9

This Class IX Mathematics (Code 041) quiz covers Unit IV: Geometry, specifically focusing on the topic “Equal Chords Equidistant from Center (Motivate)”. Test your understanding of the property and its converse. Submit your answers to check your score and download the solution PDF for offline revision.

Topic Overview: Equal Chords and Distance from Center

In the geometry of circles, the relationship between the length of chords and their distance from the center is a fundamental concept. This topic covers a key theorem and its converse, which are essential for solving problems involving chord lengths and radial distances.

1. The Property (Theorem)

Statement: Equal chords of a circle (or of congruent circles) are equidistant from the center (or centers).

This means if you have two chords, say Chord AB and Chord CD, and their lengths are equal (AB = CD), then the perpendicular distance from the center O to chord AB is exactly the same as the perpendicular distance from O to chord CD.

  • Distance Definition: The distance of a line from a point is always the length of the perpendicular drawn from the point to the line.
  • Congruent Circles: This property holds true even if the chords belong to two different circles, provided the circles have the same radius.

2. The Converse

Statement: Chords equidistant from the center of a circle are equal in length.

This is the reverse of the main theorem. If the perpendicular distance from the center to two different chords is the same, then those two chords must have the same length.

3. Calculation Formula

When solving problems, we often use the Pythagorean theorem in the right-angled triangle formed by the radius (r), the perpendicular distance from the center (d), and half of the chord length (l/2). The perpendicular from the center bisects the chord.

Formula: r squared = d squared + (l / 2) squared

Where:

  • r = Radius of the circle
  • d = Perpendicular distance from center to chord
  • l = Length of the chord

4. Important Notes

  • Longer Chords: Of any two chords of a circle, the one which is larger is nearer to the center.
  • Diameter: The diameter is the longest chord and its distance from the center is zero.
  • Bisection: Always remember that the perpendicular dropped from the center to a chord bisects the chord.

5. Extra Practice Questions

  1. If a chord of length 16 cm is at a distance of 15 cm from the center, find the radius of the circle.
  2. Two parallel chords of lengths 10 cm and 24 cm are on the same side of the center. If the radius is 13 cm, find the distance between the chords.
  3. Prove that if two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, then the chords are equal.
  4. Find the length of a chord which is at a distance of 4 cm from the center of a circle of radius 5 cm.
  5. A circle has a radius of 10 cm. Find the length of a chord that is 6 cm away from the center.