Question

The circle below is center Ed at O. Decide which length,if any, is definitely the same as the length. a) AD. b) BC. Justify your answers

Answer 1

Point O is the center of the circle.

Part (a)

`\overline{A\:\!F}` is a chord.

`\overline{OD}` is a segment of the radius and is perpendicular to
`\overline{A\:\!F}`

If a radius is perpendicular to a chord, it bisects the chord (divides the chord into two equal parts).

Therefore,
`\overline{AD}=\overline{DF}`

Part (b)

If
`\overline{BE}` was extended past point E to touch the circumference it would be a chord.

As
`\overline{OC}` is perpendicular to
`\overline{BE}`, it would bisect the chord, but as
`\overline{BE}` is only a portion of a chord,
`\overline{OC}` does not bisect
`\overline{BE}`.

Therefore, there is no length equal to
`\overline{BC}`.