Chord-arc properties, cyclic quadrilaterals, equal chords MCQs

Q1. In a circle, AB and CD are two equal chords of length 16. If the distance of AB from the centre is 6, the distance of CD from the centre is:

• A.6
• B.5
• C.8
• D.10

Answer: A. 6

Given 16, 6, asked for the unknown. By So distance of CD = 6. So distance of CD = 6. Equal chords of a circle are equidistant from the centre. So option A) 6.

Q2. In a circle, two chords AB and CD are perpendicular and intersect at P. If AP = 6, PB = 4, CP = 3, PD = 8, the radius equals:

• A.√(125)/2 = (5√5)/2
• B.5
• C.6
• D.7

Answer: A. √(125)/2 = (5√5)/2

The problem states: 6; 4; 3; 8. The unknown asked is: the unknown asked in the stem. Formula - perpendicular to AB through midpoint is x = −1. This is the equation that links the given quantities to the unknown (math, chapter 'Circles'). Substituting: perpendicular to AB through midpoint is x = −1 → perpendicular to CD through it is y = −5/2 → Centre = (−1, −5/2) → Radius² = (−1 − (−6))² + (−5/2 − 0)² = 25 + 25/4 = 125/4. Why this is the right approach - Place P at origin with AB on x-axis: A(−6,0), B(4,0). Putting it together the answer is A) √(125)/2 = (5√5)/2.

Q3. In a circle, three chords AB, CD and EF are such that AB = 10, CD = 14, EF = 10. If the perpendicular distance from the centre to AB is 5, the perpendicular distance from the centre to EF is:

• A.5
• B.7
• C.4
• D.6

Answer: A. 5