Geometrically, a figure composed of all points whose distance from a plane to a fixed point is equal to a fixed length is called a circle. A fixed point is called the center of the circle and a fixed length is called the radius.
Trajectory theory: the trajectory of a moving point on a plane with a certain length as the center is called a circle.
Set theory: The set of points whose distance to a fixed point is equal to a fixed length is called a circle.
Correlation quantity of circle
Pi: The ratio of the circumference of a circle to its diameter and length is called pi, and its value is 3.14159265358979323846 …, which is usually expressed by π. In calculation, 3. 14 16 is often taken as its approximate value.
Arc chord: the part between any two points on the circle is called arc, or simply arc. An arc larger than a semicircle is called an upper arc, and an arc smaller than a semicircle is called a lower arc. A line segment connecting any two points on a circle is called a chord. The chord passing through the center of the circle is called the diameter.
Central angle and central angle: the angle of the vertex on the center of the circle is called the central angle. The angle at which the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle.
Inner and outer center: the circle passing through the three vertices of the triangle is called the circumscribed circle of the triangle, and its center is called the outer center of the triangle. A circle tangent to all three sides of a triangle is called the inscribed circle of the triangle, and its center is called the heart.
Sector: On a circle, the figure enclosed by two radii and an arc is called a sector. The development diagram of the cone is a sector. The radius of this sector becomes the generatrix of the cone.
Letter representation of correlation between circles
Circle-⊙ radius -R arc-⌒ diameter-D.
Sector arc length/conical generatrix -l circumference -c area -s
[positional relationship between circle and other figures]
Position relationship between circle and point: Take point P and circle O as an example (let P be a point, then PO is the distance from the point to the center of the circle), where P is outside ⊙O, and PO > R；; P on ⊙O，po = r； P is within ⊙O, and po < r.
There are three positional relationships between a straight line and a circle: there is no separated common point; There are two things in common; A circle and a straight line have a unique common tangent point. This straight line is called the tangent of the circle, and this unique common point is called the tangent point. Take straight line AB and circle O as examples (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle): AB is separated from ⊙O, and po > r；; AB is tangent to ⊙O, po = r；; AB and ⊙O intersect, po < r.
There are five positional relationships between two circles: if there is nothing in common, one circle is called external separation and internal inclusion outside the other; If there is only one common point, a circle is called circumscribed by another circle and inscribed by another circle; There are two things in common called intersection. The distance between the centers of two circles is called the center distance. The radii of the two circles are R and R respectively, and R≥r, and the center distance is P: outward separation P > R+R; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; It contains p2cd (b) ab = 2cd; (C)AB < 2CD； AB = CD
3. As shown in figure (1), it is known that PA in B and B cuts ⊙O, and OP in C crosses AB, so there are () right angles * * in the figure that can be represented by letters.
a3 b . 4 c . 5d . 6
4. Given the radius ⊙O is 10cm, the chord AB ‖ CD, AB = 12 cm, CD = 16 cm, and the distance between AB and CD is ().
A.2cm B. 14cm C.2cm or 14cm D. 10cm or 20cm.
5. In a circle with a radius of 6 cm, the degree of the circumferential angle subtended by an arc with a length of 2 cm is ().
100 c . 120d . 130
6. As shown in Figure (2), if the degree of central angle ∠AOB is 100, then the degree of central angle ∠ACB is ().
100 c . 120d . 130
7. The radius ⊙ O is 20cm, the central angle ∠ AOB = 120, and AB is ⊙O string, which is equal to ().
A.25 cm2 B.50 cm2 C. 100 cm2 D.200 cm2
8. As shown in Figure (3), if the radius OA is equal to the chord AB, passes through the tangent BC⊙O of B, and BC=AB, OC intersects at E and E ⊙O, and AC intersects at D ⊙O, then the sum degrees are () respectively.
A. 15， 15 B.30， 15 C. 15，30 D.30，30
9. If the radii of two circles are r and r (r >) respectively; R), the center distance is d, R2+d2=r2+2Rd, then the positional relationship between two circles is ().
A. internal or external.
10. The generatrix of the cone is 5cm long, and the radius of the bottom surface is 3cm long, so the central angle of its side development diagram is ().
225 D.2 16
Fill in the blanks: (4 points for each small question, * * * 20 points);
1 1. A chord divides the circle into two parts, 1∶3, and the degree of the central angle subtended by the lower arc is.
12. If the diameter ⊙O is 10cm and the chord AB=6cm, the distance from the center o to the chord AB is _ _ _ _ _ cm.
13. In ⊙O, the relationship between the circumferential angles of the chord AB is _ _ _ _ _ _ _.
14. As shown in Figure (4), if in ⊙ O, AB and CD are two diameters, and the degree of the chord CE‖AB is 40, then ∠ BOD =.
(4)
15. Point A is a point outside the circle with a radius of 3, and its distance to the nearest point of the circle is 5, so the length of the tangent passing through point A is _ _ _ _ _ _ _.
16. The radius of ⊙O is 6, the chord AB of ⊙ o is 6, and the positional relationship between concentric circles with radius of 3 and straight line AB is _ _ _ _ _ _ _ _.
17. The two circles are tangent and the center distance is 10cm. If the radius of a circle is 6 cm and the radius of another circle is _ _ _ _.
18. If the degree of the arc is enlarged by 2 times and the radius is original, then the ratio of the arc length to the original arc length is _ _ _ _ _.
19. As shown in Figure (5), A is the point outside ⊙O with a radius of 2, OA=4, AB is the tangent of ⊙O, point B is the tangent point, and the chord BC ‖OA connects AC, so the area of the shaded part in the figure is _ _ _ _ _ _ _ _.
20. As shown in Figure (6), it is known that the central angle of the fan-shaped AOB is 60 and the radius is 6, and c and d are bisectors respectively, so the area of the shaded part is equal to _ _ _ _ _ _.
III. Answer questions (2 1~23 questions, 8 points for each question, 24~26 questions, 12 questions, 60 points for each question).
2 1. As shown in the figure, in two concentric circles with O as the center, the chord AB of the big circle intersects the small circle at points C and D.
Try to explain: AC=BD.
22. As shown in the figure, in Rt△ABC ∠ BAC = 90, AC = AB = 2, and the circle with the diameter of AB intersects BC in D, find the area of the shadow part of the figure.
23. As shown in the figure, AB is the diameter of ⊙O, AE bisects ∠BAC and intersects with ⊙O at point E, and intersects with the tangent of ⊙O at point D. Try to judge the shape of△ △AED and explain the reasons.
24. As shown in the figure, there is a circular arch bridge with a span of 60m and an arch height of18m. When the flood reaches a span of only 30 meters, emergency measures should be taken. If the vault is only 4 meters away from the water surface, that is, PN=4 meters, should emergency measures be taken?
25. As shown in the figure, the quadrilateral ABCD is inscribed with a semicircle O, and AB is the diameter. (1) Please add a condition to make the quadrangle ABCD in the figure an isosceles trapezoid. This condition is (only fill in one condition). (2) If CD= AB, please design a scheme to divide the isosceles trapezoid ABCD into three parts with equal area and prove it.
26. Take a point A on the ray OA, make OA = 4cm, and make a circle with a diameter of 4cm with A as the center. Q: When the ray OB passes through the acute angle α of O, OA and OB are separated (1); (2) Tangency; (3) intersection.