Chords of a Circle

A chord is a line segment whose endpoints lie on the circle. The most fundamental theorem you must master for board exams states that a perpendicular from the center of a circle to a chord bisects the chord. Conversely, the line drawn from the center of a circle to the midpoint of a chord is perpendicular to the chord. Examiners frequently use this concept in numerical problems to find the length of a chord or the distance from the center by applying the Pythagorean theorem to the right-angled triangle formed by the radius, the perpendicular distance, and half the chord.

Another critical property frequently tested in both short and long questions involves the equidistant nature of chords. Congruent chords of a circle are equidistant from the center, and conversely, chords that are equidistant from the center are congruent. You should also remember that equal chords subtend equal angles at the center of the circle. These properties form the backbone of many analytical proofs and deductive geometry questions in BISE board papers.

For circles passing through points, a vital theorem states that one and only one circle can pass through three non-collinear points. This is the foundational principle used when you are asked to circumscribe a circle about a given triangle or find the center of a given circle arc. Make sure to memorize all standard theorems, their given/to-prove statements, and constructions, as geometry theorems carry heavy weightage in the subjective section of the Math paper.