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Given that the length of a chord is equal to the radius r, find: (1) The length of the inferior arc to which the string is directed; (2) The area of the bow formed by this chord and minor arc
(1) As shown in the figure, if the chord ab of ⊙ o with radius is r, then ⊙ OAB is an equilateral triangle, so ∠ AOB = π
3, then the minor arc of chord AB is π
3r．… (3 points)
(2) Because s △ AOB = 1
2•OA•OBsin∠AOB＝
Three
4r2，
S sector AOB = 1
2|α|r2＝1
2×π
3×r2＝π
6r2
So s arch = s sector aob-s △ AOB = (π
6−
Three
4）r2… (8 points)
The radius of circle O is 2, the distance from point P in circle O to center O is 1. The chord AB passing through point P and inferior arc AB form an arch. The minimum area of this arch is______ .
According to the meaning of the title, draw the corresponding figure, as shown in the figure: when the string ab ⊥ OP is obtained from the graph, the area of arch AB is the smallest, ? ab ⊥ OP, ? apo = 90 degrees, ? in the right triangle AOP, OA = 2, Op = 1, OAP = 30 °, AP = oa2 − op2 = 3, and ? op ⊥ AB, ᙽ AB = 2AP = 23, similarly ? OBP = 30 degrees
Given that the length of a chord is equal to the radius r, find: (1) The length of the inferior arc to which the string is directed; (2) The area of the bow formed by this chord and minor arc
(1) As shown in the figure, if the chord ab of ⊙ o with radius is r, then ⊙ OAB is an equilateral triangle, so ∠ AOB = π
3, then the minor arc of chord AB is π
3r．… (3 points)
(2) Because s △ AOB = 1
2•OA•OBsin∠AOB＝
Three
4r2，
S sector AOB = 1
2|α|r2＝1
2×π
3×r2＝π
6r2
So s arch = s sector aob-s △ AOB = (π
6−
Three
4）r2… (8 points)
If the length of a chord in a circle is equal to its radius r, then the area of the arch formed by the chord and minor arc is equal to?
According to the formula of sector area, the area of the sector is S1 = π R ^ 2 * 60 ° / 360 ° = π R ^ 2 / 6. The area of equilateral triangle composed of chord and radius of both sides is S2 = R ^ 2 * √ 3 / 4. Therefore, the area of the arch formed by this chord and minor arc is s = S1 -
In the circle with radius 6, find the arcuate area enclosed by a chord of length 6 and its inferior arc
Let the center of the circle be o, the chord AB, and the midpoint C of ab
Then OC ⊥ AB, then ⊥ OCB is a right triangle of ﹣ BOC = ∠ AOC = 30 degrees
The center angle of AB is π / 3
So the inferior arc AB = radius * π / 3 = 2 π
In the rectangular coordinate system, the center of ⊙ o is at the origin, the radius is 3, and the coordinates of center a of ⊙ a are (− 0 If the radius is 1, then the position relationship between ⊙ O and ⊙ A is______ .
According to the meaning of the title, we can get
O（0，0），|OA|=
(0+
3)2+(0−1)2＝
4=2，
∴R-r=3-1=2=|OA|，
The two circles are inscribed
As shown in the figure, it is known that O is a point on the diagonal AC of the square ABCD. With o as the center of the circle and the length of OA as the radius, ⊙ O and BC are tangent to M As shown in the figure, O is a point on the diagonal of the square ABCD. The circle with o as the center and OA length as the radius is tangent to m and AB, and ad intersects EF respectively (2) If the side length of the square is 1, find the o radius of the circle
solution
Let the radius OA = X
Then om = X
In triangle OCM
OC=√2*X
AC = √ 2
OA=√2-√2*X
X=√2-√2*X
solve equations
X=2-√2
As shown in the figure, in the rectangular coordinate system, the ⊙ o with the coordinate origin as the center and radius of 1 intersects with X axis at a and B points, and Y axis intersects with C and D points. E is a point on the first quadrant of ⊙ o, and the line BF intersects ⊙ o at point F, and ∠ ABF = ∠ AEC, then the function expression of line BF is______ .
According to the circular angle theorem, ∠ AEC = 1
2∠AOC=45°，
∵∠ABF=∠AEC=45°，
The point F coincides with point C or D;
When point F coincides with point C, let the analytic formula of line BF y = KX + B,
be
b＝1
K + B = 0
k＝−1
b＝1
The analytical formula of the line BF is y = - x + 1,
When point F coincides with point D, y = X-1 can be obtained
In rectangular coordinate system, the radius of circle O is 5, and the coordinates of center o are (- 1, - 4). Try to judge the position relationship between point (3, - 1) and circle o
The distance between points (- 1, - 4) and points (3, - 1)
√ (4 ^ 2 + 3 ^ 2) = √ 25 = 5 = radius of circle O,
So, the point (3, - 1) is on the circle o
In a rectangular coordinate system, a point whose abscissa and ordinate are integers is called a lattice point. Given that the center of a circle is at the origin and its radius is equal to 5, then the grid point on the circle has______ One
According to Pythagorean theorem, there are 4 points with a distance of 5 from the center of the circle on the coordinate axis. According to the Pythagorean theorem, there are 8 points with a distance of 5 from the center of the circle, a total of 12 points, as shown in the figure
RELATED INFORMATIONS
- 1. In a rectangular coordinate system, a point whose abscissa and ordinate are integers is called a lattice point. Given that the center of a circle is at the origin and its radius is equal to 5, then the grid point on the circle has______ One
- 2. As shown in the figure, in the plane rectangular coordinate system, the coordinates of the three vertices of △ ABC are a (2, 3), B (2, 1), C (3, 2) (1) Judge the shape of △ ABC; (2) If △ ABC is rotated one circle along the line of edge AC, the volume of the rotating body is obtained
- 3. As shown in the figure, it is known that in the rectangular coordinate system xoy, points a and C are on the positive half axis of the x-axis and y-axis respectively, and the point O is at the coordinate origin. The right angle vertex o of the isosceles right angle triangle OEF is at the origin, e and F are on OA and OC respectively, and OA = 4, OE= 2. Rotate the triangle plate OEF anticlockwise around point O to oe1f1, and connect CF1 and AE1 (1) Verification: △ oae1 ≌ △ ocf1; (2) If the triangle plate OEF rotates anticlockwise around point O, is there a position that makes OE ∥ CF? If yes, request the coordinates of e point; if not, please explain the reason
- 4. In the plane rectangular coordinate system, the coordinate of point a is (4,0), the point P is on the straight line y = - x + m, and AP = OP = 4
- 5. It is known that O is the origin of the plane rectangular coordinate system. The straight line passing through the point m (- 2,0) and the circle x ^ 2 + y ^ 2 = 1 intersect P and Q If the vector OP * vector OQ = - 1 / 2, find the equation of the line L If the area of the triangle OMP and the triangle OPQ are equal, find the slope of the line L
- 6. As shown in the figure, in the plane rectangular coordinate system, point O is the origin, the diagonal ob of diamond oabc is on the X axis, and the vertex A is in the inverse scale function y = 2 On the image of X, the area of the diamond is______ .
- 7. As shown in the figure, in the plane rectangular coordinate system, the image of quadratic function intersects C at Y axis, a and B at x axis, points a and B are on both sides of the origin The coordinate of a is (- 3,0), Ao = co = 3BO (1) Find this quadratic function expression (2) If the point d (- 2, y) is a point on the parabola, find the area of △ BCD (3) In (2), if point P is a moving point between B and D on the parabola, when P moves to what position, the area of △ BDP is the largest? Find the coordinates of point P and the maximum area of △ BDP!
- 8. In the plane rectangular coordinate system, O is the coordinate origin. The point P (m, n) is on the image of the inverse scale function y = K / X (1) If M = k, n = K-2, find the value of K (2) If M + n = radical 2 K, Op = 2, and the inverse proportional function y = K / X satisfies: when x > 0, y decreases with the increase of X, and the value of K is calculated
- 9. In the plane rectangular coordinate system, the points a (- 4,0) and B (2,0) are known. If the point C is in the first order function y = - 1 On the image of 2x + 2, and △ ABC is a right triangle, then the point C satisfying the condition has () A. 1 B. 2 C. Three D. Four
- 10. It is known, as shown in the figure: in the plane rectangular coordinate system, O is the coordinate origin, the quadrilateral oabc is a rectangle, the coordinates of points a and C are a (10, 0), C (0, 4), respectively; point D is the midpoint of OA, and point P moves on the edge of BC; when △ ODP is an isosceles triangle with waist length of 5, the coordinates of point P are______ .
- 11. Given that ⊙ O1 and ⊙ O2 are circumscribed at point a, the radius of ⊙ O1 is r = 2, and the radius of ⊙ O2 is r = 1, then the circle with radius 4 has () A. 2 B. Four C. Five D. 6
- 12. The function is equal to the root x + 4, and the value range of the independent variable x is
- 13. It is known that m (x, y) is a set of inequalities where x is greater than or equal to 0, less than or equal to root 2, y is less than or equal to 2, and X is less than or equal to root sign 2Y If point a (root 2,1), then z = vector OM, the maximum value of point multiplication vector OA is
- 14. It is known that the distance from point a (2m + 1,3m-5) to the x-axis is twice the distance from the point to the y-axis
- 15. If the chord length of the straight line 2x-y + 1 = 0 and the section circle x? 2 + y? 2 = R? 2 is equal to 5, find the radius of the circle
- 16. In the plane rectangular coordinate system, the distance from the point P (- 3, - 4) to the X axis is () and the distance from the point to the Y axis is
- 17. In the plane rectangular coordinate system xoy, let the image of quadratic function f (x) = x ^ 2 + 2x + B have three intersections with the two coordinate axes, and the circle passing through these three intersections is marked as C (I) find the equation of circle C; (II) set point a is a fixed point passed by circle C (its coordinates are independent of B). Ask whether there is a constant k such that the straight line y = KX + K and circle C intersect at points m, N, and | am | = | an |. If it exists, find the value of K; if not, please explain the reason
- 18. The coordinates of the center P of a circle (- 2,5), and its radius is 6, then the origin of the coordinates is at point P
- 19. How to deal with such problems as how many times the radius of a circle, how many times its diameter, how many times its circumference and how many times its area? I will have the final exam tomorrow. Is there any formula?
- 20. When the radius of a circle is several centimeters, its circumference and area are equal