2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 20 Hard

Let ABCD be a rhombus with ∠ADC=46^{\circ} . Let E be the midpoint of \overline{CD}, and let F be the point on \overline{BE} such that \overline{AF} is perpendicular to \overline{BE}. What is the degree measure of ∠BFC?

  • A.

    110

  • B.

    111

  • C.

    112

  • D.

    113

  • E.

    114

Answer:D

​

Refer to the figure. AF \perp BE and BO \perp AC. Denote the intersection between AC and BD as H. Here OH is the orthocenter of \triangle ABG.

\angle A F O=\angle A B O=23^{\circ}

\angle O F E=90^{\circ}-\angle AFO=90^{\circ}-23^{\circ}=67^{\circ}.

∵ \angle ACD=67^{\circ},

∴ OFCE are concyclic,

∴ \angle CFE=\angle COE=90^{\circ}-23^{\circ}=67^{\circ},

∴ \angle BFC=180^{\circ}-67^{\circ}=113^{\circ}.

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