Who has a common knowledge point about junior high school round 10

the radius perpendicular to the tangent point; A straight line that passes through one end of the radius and is perpendicular to this radius is the tangent of the circle.

Tangent is determined by a straight line that passes through the outer end of the radius and is perpendicular to this radius as a tangent of a circle.

Properties of tangents: (1) A straight line passing through the tangent perpendicular to the radius of the tangent is a tangent of a circle. (2) A straight line perpendicular to the tangent must pass through the center of the circle. (3) The tangent of the circle is perpendicular to the radius passing through the tangent point.

Tangent length theorem: The length of the two tangents from the outer point of the circle to the circle is equal, and the point is bisected with the line at the center of the circle.

The tangent line theorem of a circle intersects a tangent line with a secant line at point p, the tangent line intersects at point c, and the secant line intersects at two points a b, then there is pc 2=pa·pb

The secant theorem is similar to the cut line theorem Two secant lines intersect at point p, secant m intersects at two points A1 B1, and secant n intersects at two points A2 B2.

then PA1·PB1=PA2·PB2

Properties and theorems about circumferential and central angles.

In the same circle or equal circle, if one set of quantities in two central angles, two circumferential angles, two sets of arcs, two strings, and two chord center distances are equal, then the rest of the groups of quantities corresponding to them are equal.

The circumferential angle of an arc is equal to half of the central angle of the circle it opposes.

The circumferential angle of the diameter is a right angle. The chord to which the circumferential angle of 90 degrees is aligned is the diameter.

The formula for calculating the central angle of the circle: =(l 2 r) 360°=180°l r=l r(radian).

That is, the degree of the central angle of the circle is equal to the degree of the arc to which it is opposed; The degree of the circumferential angle is equal to half the degree of the arc it is against.

If an arc is twice as long as another arc, then the circumferential and central angles of the arc are twice as long as the other arc.

Properties and theorems about circumscribed and inscribed circles.

A triangle has a uniquely determined circumscribed circle and an inscribed circle. The center of the circumscribed circle is the intersection of the perpendicular bisector of each side of the triangle, and the distance to the three vertices of the triangle is equal;

The center of the inscribed circle is the intersection of the bisector of each inner corner of the triangle, and the distance to the three sides of the triangle is equal.

r=2s l(r: radius of the inscribed circle, s: area of the triangle, l: perimeter of the triangle).

A tangent line of two tangent circles crosses the tangent point (a central line: a straight line in which the centers of two circles are connected).

The midpoint m of the chord pq in the circle O, the crossing point m is either the two strings ab, cd, and the chord AD and BC intersect pq at x and y respectively, then m is the midpoint of xy.

4) If two circles intersect, then the line segment connecting the centers of the two circles (a straight line can also be) bisects the common chord perpendicularly.

5) The degree of the chord tangent angle is equal to half the degree of the arc it clamps.

6) The degrees of the inner angle of the circle are equal to half of the sum of the degrees of the arc to which the angle is opposed.

7) The degree of the outer angle of the circle is equal to half of the difference between the degrees of the two arcs truncated by this angle.

8) The circumference is equal, and the area of the circle is larger than that of a rectangle, square, or triangle.

All the knowledge points of the junior high school circle:

1. The basic properties and theorems of circles.

1) The circle is an axisymmetric figure, and its axis of symmetry is any straight line that passes through the center of the circle. A circle is also a center-symmetrical figure, and its center of symmetry is the center of the circle.

2) In the same circle or equal circle, if one set of quantities in 2 central angles, 2 circumferential angles, 2 arcs, and 2 strings are equal, then the other groups of quantities corresponding to them are equal.

3) The tangent of the circle is perpendicular to the diameter of the tangent point; A straight line that passes through one end of the diameter and is perpendicular to this diameter is the tangent of the circle.

2. Definition of circles and the correlated quantities of circles.

1) There are three positional relationships between straight lines and circles: no common point is separated; There are 2 common points for intersection; A circle and a straight line have a single common point that is tangent, this straight line is called the tangent of the circle, and this only common point is called the tangent point.

2) On a circle, a figure surrounded by two radii and an arc is called a fan. The side view of the conic is a fan. The radius of this sector becomes the busbar of the cone.

3) There are 5 positional relationships between two circles: if there is no common point, one circle is called outside the other circle, and it is called contained; If there is a single common point, a circle outside another circle is called an external incision, and an internal incision is called an internal incision; There are 2 common points called intersections. The distance between the centers of two circles is called the center distance.

The knowledge points of the first three circles are summarized as follows:

1. Definition of a circle.

1) In a plane, the shape formed by the rotation of the line segment oa around one of its endpoints o and the rotation of the other endpoint a is called a circle. The fixed endpoint o is called the center of the circle, and the line segment oa is called the radius, as shown in the figure on the right.

2) A circle can be seen as a set of points whose distance from the plane to the fixed point is equal to the fixed length, the fixed point is the center of the circle, and the fixed length is the radius of the circle.

Note: The position of the circle is determined by the center of the circle, the size of the circle is determined by the radius, and two circles with equal radius are equal circles.

2. The concept of circle.

1) String: A line segment that connects any two points on a circle. (As shown in the CD on the right).

2) Diameter: The string that passes through the center of the circle (as ab in the figure on the right). The diameter is equal to 2 times the radius.

3) Arc: The part between any two points on a circle is called an arc. (Such as CD, CAD in the figure on the right) in which the arc larger than the semicircle is called the superior arc, such as CAD, the arc smaller than the semicircle is called the inferior arc.

4) Center Angle: As shown in the figure on the right, COD is the center angle.

3. The relationship between the central angle, arc, chord, and chord centroid distance.

1) Theorem: In the same circle or equal circle, the arcs of the opposite central angles of the equal circles are equal, and the chord centroid distances of the paired strings are equal.

2) Corollary: In the same circle or equal circle, if one set of quantities is equal in the central distance of two circles, two arcs, two strings, or two strings, then the rest of the groups of quantities corresponding to them are equal.

4. A circle of three points.

1) Theorem: Three points that are not on the same straight line determine a circle.

2) The circumscribed center of the triangle (outer center) is the intersection of the three perpendicular bisectors.

5. Perpendicular diameter theorem.

Bisect the string perpendicular to the diameter of the string, and bisect the two arcs opposite the string. Corollary:

1) The diameter of the bisector chord (not the diameter) is perpendicular to the chord, and the two arcs of the bisector chord are opposed;

The perpendicular bisector of the string passes through the center of the circle, and bisects the two arcs opposite the chord;

The diameter of one string to which the bisector string is paired, the perpendicular bisector of the chord, and the other arc to which the bisector chord is paired.

2) The arcs sandwiched by the two parallel chords of the circle are equal.

1. In a plane, the closed curve formed by rotating around a point and at a distance of a certain length is called a circle.

2. A circle has an infinite number of axes of symmetry.

3. The circle is a kind of conic curve, which is obtained by a flat truncated cone parallel to the bottom surface of the cone.

4. The circle is prescribed as 360°, which is when the ancient Babylonians observed the rising sun on the horizon, moving a position about every 4 minutes, and 360 positions in 24 hours a day, so the inner angle of a circle was prescribed to be 360°. This ° represents the sun.

Formulas related to circles

1. The area of the semicircle: s semicircle = (r 2) 2. (r is the radius).

2. The area of the ring: S big circle - S small circle = (r 2-r 2) (r is the radius of the large circle, r is the radius of the small circle).

3. The circumference of the circle: orange with c=2 r or c=d. (d is the diameter, r is the radius).

4. The circumference of the semicircle: d+(d) 2 or d+ r. (d is the diameter, r is the radius).

5. The arc length of the sector l=the central angle (radian) r= n r 180 (the central angle) (r is the radius of the fan).

6. The sector area s=n r 360=lr 2 (l is the arc length of the fan).

7. The radius of the bottom surface of the cone r=nr 360 (r is the radius of the bottom surface) (blind and dry n is the central angle of the circle).

is the sum of an infinite number of small sector areas, so in the last formula, the sum of the small arcs of the segments is the circumference of the circle 2 r, so there is s = r.

The knowledge points of the junior high school circle are as follows:

1. The outside of the circle can be seen as a collection of points where the distance of the center of the circle is greater than the radius.

2. The theorem that the diagonal complementarity of the inner quadrilateral beat of the circle is complementary, and any outer angle is equal to its inner diagonal.

3. The tangent length theorem leads two tangent lines from a point outside the circle, their tangent lengths are equal, and the center of the circle and the line of this point divide the angle between the two tangents.

4. The tangent of the circle through each subnox, and the polygon with the intersection of the adjacent tangents as the vertex is the circumscribed regular n-sided of the circle.

5. The arc length formula l=a*ra is the radian degree of the central angle r>0 sector area formula s=1 2*l*r.