Given a semicircle with O as the centre, as shown in the figure, the ratio (AC + CB) / AB is _______, where AC, CB and AB are chords. A. sqrt(2) B. sqrt(3) C. 2 D. 3

GATE 2020 · General Aptitude · Circle · medium

Answer: The ratio (AC + CB) / AB = sqrt(2).

1. Set up coordinates: With radius r: A = (-r, 0), B = (r, 0), O = (0, 0), C = (0, r) (apex of semicircle, midpoint of arc as shown in figure).
2. Compute chord AC: AC = sqrt((0-(-r))^2 + (r-0)^2) = sqrt(r^2 + r^2) = r*sqrt(2).
3. Compute chord AB: AB = distance from A=(-r,0) to B=(r,0) = 2r.
4. Compute the ratio: (AC + CB) / AB = (r*sqrt(2) + r*sqrt(2)) / (2r) = 2r*sqrt(2) / (2r) = sqrt(2).