The chords AB and CD of the circle meet at point E. The ratio CE: ED = 2: 3. Find ED if AB = 20 cm and AE is 3 cm less than BE.

univerkov | February 16, 2021 | education

Let the segment BE = X cm, then, by condition, AE = (X – 3).

AE + BE = 20.

X – 3 + X = 20.

2 * X = 23.

X = 23/2 = 11.5 cm.

BE = 11.5 cm.

AE = 11.5 – 3 = 8.5 cm.

By condition, CE / DE = 2/3.

2 * DE = 3 * CE

By the property of chords intersecting at one point, the product of the segments formed at the intersection of one chord is equal to the product of the segments of the other chord.

AE * BE = CE * DE.

8.5 * 11.5 = CE * DE.

Multiply one part of the equality by 2.

195.5 = CE * 3 * CE = 3 * CE: 2.

CE: 2 = 195.5 / 3 = 65.17.

CE = 8.07 cm.

ED = 3 * CE / 2 = 12.1 cm

Answer: ED = 12.1 cm.

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