Circle — Important Questions

The perpendicular from center to chord bisects the chord, Equal chords are equidistant from center, All radii of a circle are equal

Statement 1: Theorem 23 states this. Statement 2: Theorem 24 proves this. Statement 4: By definition of circle. Statement 3 is wrong as diameter is the longest chord. Statement 5 is wrong as no chord can exceed diameter length.

10 cm

Let distances from center be x and (14-x). Half-lengths: 8 cm and 6 cm. From Pythagorean theorem: r² = x² + 8² and r² = (14-x)² + 6². Solving: x² + 64 = (14-x)² + 36. x = 6. So r² = 36 + 64 = 100, r = 10 cm.

They are equidistant from center, They subtend equal angles at center, They cut equal arcs

Equal chords have three main properties: (1) Equidistant from center (Theorem 24), (2) Subtend equal central angles, (3) Cut equal arcs. They don't need to pass through center or be parallel.

AB = CD

In a circle, equal arcs cut equal chords. This is a fundamental property: if arc AB = arc CD, then chord AB = chord CD.