Question

Chords AB and CD of a circle intersects each other in point M. The center of the circle is P. The radius of the circle is 13 cm and PM= 5 cm. Find the product $CM\times DM.$

Answer 1

Intersecting chords theorem states that when two chords intersect each other inscale a circle, the product of their segments are equal also, diameter is the longest chord of the circle also now, draw a diameter of the circle passing through points P and M. it is the line XY as shown in the attachment

Length $XY=2\ast$ radius $=2\ast 13=26cm$

$MY=PY-PM=13-5=7cm$

$XM=XP+PM=13+5=18cm$

By the intersecting chords theorem we have

$CM\ast PM=XM\ast YM=18\ast 7=126$

$\therefore CM\ast PM=126$