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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

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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 $$\overline{BC}$$.
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#a