Use similar triangles.

△OPB and △OQC are similar

(If you need the proof let me know)


If PQ = y cm, then OP = (24 - y)cm

Since they are similar,

OP/OQ = PB/QC
(24 - y)/24 = 6/12 = 12/24

So 24 - y = 12
And y = 24 - 12 = 12

THANKS for your help but we also need the proof so sorry...

angle POB = angle QOC (common angle)

angle OQC = 90° since height of cone OQ is perpendicular to base CD


O,P,Q are collinear from the diagram.


OQ passes through P. Since P is the midpoint of AB , and also the centre of the hemisphere, OQ passes through the centre and bisects AB

OQ is therefore perpendicular to AB.

(The perpendicular bisector of a chord passes through the centre. AB happens to be the diameter of the hemisphere, and to diameter is effectively a chord that passes through the centre)


Since OQ is perpendicular to both AB and CD, AB and CD are parallel.

So angle OPB = OQC = 90°
(corresponding angles, AB // CD)

By AA similarity, triangle OPB and OQC are similar

Thank you so much!!