Contributed by Liu
First, multiple-choice questions (3 points for each question, ***30 points)
1. Given that the radius of a circle is, and the distance from the center of a circle to a straight line is, then the number of points shared by this straight line and this circle is ().
A.0b.1c.2d. Not sure.
2. It is known that the radii of two circles are 5 and 2 respectively, and the center distance is 3, so the positional relationship between the two circles is ().
A. containing inscribed C. intersecting D. circumscribed
3. In the figure below, it is both an axisymmetric figure and a centrally symmetric figure ().
A. isosceles trapezoid B. parallelogram C. equilateral triangle D. circle
4. As shown in the figure, ⊙O is circumscribed by △ABC, AD is the diameter of ⊙O, and ∠ ABC = 30, then ∠CAD degree ().
30 BC to 40 BC
5. In 5.RT △ ABC, ∠ c = 90, AB=5, and the radius of inscribed circle is 1, then the perimeter of the triangle is ().
15， 12， 13， 14
6. The correct proposition in the following propositions is ()
The straight line perpendicular to the radius of a circle is the tangent of the circle.
The straight line passing through the outer end of the radius is the tangent of the circle.
A straight line whose distance from a point to the center of the circle is equal to the radius of the circle is the tangent of the circle.
D the straight line whose distance from the center of the circle is equal to the radius of the circle is the tangent of the circle.
7. In ⊙O with a radius of 1, if the chord AB = 1, the length of the arc AB is ().
A.B. C. D。
8. If the central angle of the sector is 150 and the sector area is 240cm2, then the arc length of the sector is ().
A.5cm cm B.10cm C.20cm D.40cm.
9. If the radius ⊙O is 3cm, the point M is the point outside ⊙ o, and OM=4cm, the radius of the circle tangent to ⊙ o with m as the center is ().
A. 1 cm B.7 cm C. 1 cm or 7 cm D. Not sure.
10. The following statement is true ()
(1) An arc in which two endpoints can overlap is an equal arc.
(2) Any chord of a circle must divide the circle into two parts: the lower arc and the upper arc.
(3) Any three points on the crossing plane can make a circle.
(4) Any circle has one and only one inscribed triangle.
(5) The distance from the outer center of the triangle to each vertex is equal.
1。
Fill in the blanks (3 points for each question, 30 points for * * *)
1 1. If there is a point P outside the ⊙O with a radius of 5cm, then the ⊙P tangent to ⊙O can be drawn as _ _ _ _ _ _.
12. point p is within ⊙O, and OP = 2 cm. If the radius ⊙O is 3 cm, the length of the shortest chord passing through point P is
13. As shown in the figure, in ⊙O, AB is the diameter, and in D, the bisector of ∠ACB intersects ⊙O, then ∠ Abd =
14. In a circle with a radius of 5 cm, the distance from the center of the circle to a chord with a length of 8 cm is
15. The equilateral triangle ABC with a side length of 2 is inscribed in ⊙O, so the distance from the center of O to the side of △ABC is
16. As shown in the figure, AB and AC are two tangents of ⊙O, and the tangents are B and C respectively, and D is a point on the optimal arc (BC, known as ∠ BAC = 80, then ∠ BDC =.
17. The radius ratio of two circles is 5:3, the center distance is 32cm when circumscribed and _ _ _ _ _ _ cm when inscribed.
18. If the radii of two circles are r and () respectively, and the distance between the centers is sum, the positional relationship between the two circles is
19. The radii of ⊙O and ⊙ o are 8 and 5 respectively, and there is no common point between the two circles, so the value range of the center distance OO is
20. It is known that every two circles in the figure are tangent, and the radius ⊙O is 2R, the radius ⊙O 1 and the radius ⊙O2 is r, then the radius ⊙O3 is
Iii. Answering questions (***40 points)
2 1.(8 points) As shown in the figure, it is known that the chord AB is equal to the radius, connecting OB and extending BC = OB.
What is the relationship between (1)AC and ⊙O?
(2) Please find a little D on ⊙O to make AD=AC (finish drawing by yourself and prove your conclusion).
22.(8 points) As shown in the figure, it is known that the radius of fan-shaped AOB is 12, OA⊥OB, C is a point on OB, and the semicircle O 1 with the diameter of OA is tangent to the semicircle O2 with the diameter of BC at point D, and the area of the shaded part in the figure is found.
23.(8 points) As shown in the figure, draw two tangents PA, PB from point P to ⊙ O, with tangents A, B and AC as chords and BC as the diameter of ⊙ O. If ∠ P = 60 and PB=2cm, find the length of AC.
24.(8 points) Two soap bubbles with the same size are stuck together, and their cross sections are as shown in the figure (O point, O' as the center). The soap film PQ separating two soap bubbles is on a straight line, and TP and NP are the tangents of two circles respectively. Find the size of ∠TPN.
25.(8 points) As shown in the figure, the diameter AB of the semicircle O is known, and the right-angle vertex of the triangular plate is fixed on the center O. When the triangular plate rotates around the point O, the two right-angle edges of the triangular plate intersect with the circumference of the semicircle at the points C and D respectively, and the connection points AD and BC intersect at the point E. Is the line segment BD always equal to DE? If yes, please prove it; If not, please explain why.
primary work