Question

In circle K shown, secant PEC intersects chord BKD at F. If AE = AB, mED = 32, and mBC mCD = 2 :1, then find the following:

(a) measurement of angle P

(b) measurement of angle CFD

(c) Is AE perpendicular to PEC at E? This diagram is not to scale. Provide evidence to support your answer.

Answer 1

(a) ∠P = 60°

(b) ∠CFD = 104°

(c) no

Explanation:

It is convenient to start by finding the measures of the arcs around the circle. We know BD is a diameter, so arc BD measures 180°. Point C divides it into parts with the ratio 2:1, so ...

arc BC = (2/3)(180°) = 120°

arc CD = (1/3)(180°) = 60°

Arc BE is the supplement to arc ED, so is ...

arc BE = 180° -32° = 148°

and arcs BA and AE are each half that, so are ...

arc BA = arc AE = 148°/2 = 74°

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(a) Arc ABC = arc BA +arc BC = 74° +120° = 194°

Arc AE = 74° (from above). The angle at P is half the difference of arcs ABC and AE:

∠P = (194° -74°)/2

∠P = 60°

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(b) ∠CFD is half the sum of arcs CD and BE:

∠CFD = (60° +148°)/2

∠CFD = 104°

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(c) ∠CEA is half the measure of arc CBA, so is ...

∠CEA = 194°/2

∠CEA = 97°

AE is not perpendicular to PC.

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The attached diagram is to scale.