Chords AB and CD intersect at point E. Find the value of CD if AE = 4cm, BE = 9cm, and the length of CE

Chords AB and CD intersect at point E. Find the value of CD if AE = 4cm, BE = 9cm, and the length of CE is four times the length of DE.

Consider triangles BEC and AED in which the angles BEC and AED are equal as the vertical angles at the intersection of the chords AB and BC. The inscribed angles ABC and ADC are based on one arc AC, then these angles are equal, and therefore the triangles BEC and AED are similar in two angles.

Let the length of the segment DE = X cm, then the length of the segment CE = 4 * X cm.

In similar triangles ВЕС and AЕD:

DE / AE = BE / CE.

X / 4 = 9/4 * X.

4 * X2 = 4 * 9 = 36

X2 = 9.

X = DE = 3 cm.

Then CE = 4 * 3 = 12 m.

CD = DE + CE = 3 + 12 = 15 cm.

Answer: The length of the CD segment is 15 cm.