This is the 2022 high school entrance examination real question
I saw on Toutiao today. As shown in the figure, AB is the diameter of ⊙O. Rotate the chord AC 30 degrees clockwise around point A to get AD. At this time, the corresponding point D of point C falls on AB. on, extend CD to intersect ⊙O at point E. If CE=4, the area of the shaded part in the figure is ____.
High School Entrance Exam Geometry Question
First review the question: We need to find out those secondary known conditions from the given known conditions condition. For example, △ADC is an isosceles triangle with two base angles of 75°; the circumferential angle subtended by the diameter is 90°. The circumferential angle subtended by
semicircle is 90°. The AE arc and the BE arc are exactly semicircles, and the sum of the circumferential angles they subtend is 90°, that is, ∠C+∠BAE=90°.
∵∠C=75°, ∴∠BAE=15°, ∠CAE=45°.
is much simpler now. The question is to ask for the area enclosed by the chord corresponding to the 45° circumferential angle and the minor arc of the circle. The chord length is known to be 4.
45° circumferential angle corresponds to 90° central angle. △OCE is an isosceles right triangle . In this way, the radius of the circle can be found to be 2√2. The area of the shaded part of
is the area of the 90° sector minus the area of the isosceles right triangle.
sector area=πr²/4,
area of isosceles right triangle=r²/2,
shaded area=πr²/4-r²/2=(π-2)r²/4=2(π-2).
To summarize: This question is a basic concept question, testing whether students are proficient in the properties of circles. Other knowledge points include the area of an isosceles right triangle and a sector, which are all basic knowledge.
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