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As shown in the figure, the radius of ⊙ o is 3cm, point B is a point outside ⊙ o, OB intersects ⊙ o at point a, and ab = OA, starting from point a, the moving point P moves anticlockwise on ⊙ o at the speed of π cm / s, returns to point a and stops immediately A. 1 B. 5 C. 0.5 or 5.5 D. 1 or 5
Connect OP,
∵ the line BP is tangent to ⊙ o,
∴OPB=90°，
∵AB=OA=OP，
∴OB=2OP，
∴∠PBO=30°，
∴POB=60°，
The length of the arc AP is 60 π· 3
180=π，
That is, the time is π △ π = 1 (second);
When point P ', the line BP is tangent to ⊙ o,
The length of the arc app 'is (360 − 60) π· 3
180=5π，
That is, the time is 5 π - π = 5 (seconds);
Therefore, D
As shown in the figure, AB cuts ⊙ o in B, OA intersects ⊙ o in C, ∠ a = 30 °, if ⊙ o radius is 3cm, find the length of Ao
solution
: connect ob, as shown in the figure,
∵ AB cut ⊙ o in B,
∴OB⊥AB，
∴∠ABO=90°，
In RT △ ABO, ∵ a = 30 ° ob = 3cm,
∴OA=2OB=6cm．
Let ⊙ be the length of the point ⊙ o ⊙ o = 0cm
Connect OC;
∵ AB and ⊙ o are tangent to point C,
∴OC⊥AB，
∵OA=OB，
∴AC=BC=5，
In RT △ AOC,
OA＝
AC2+OC2＝
52+42＝
41（cm）．
A: what is the advantage of OA
41cm．
As shown in the figure, AB and ⊙ o are tangent to point C, OA = ob (1) As shown in Fig. 1, if the diameter of ⊙ o is 8cm and ab = 10cm, calculate the length of OA (the root number is retained in the result); (2) As shown in Fig. 2, OA, OB and ⊙ o intersect points D and e respectively, and connect CD and CE. If the quadrilateral odce is diamond shaped, calculate OD OA value
(1) In RT △ ACO, OC = 12 × 8cm = 4cm, AC = 5cm. According to the Pythagorean theorem, OA = ac2 + oc2 = 41 (CM); (2) ∵ odce of quadrilateral is diamond,
As shown in the figure, given that the radius of ⊙ o is 1, AB and ⊙ o are tangent to point a, OB and ⊙ o intersect at point C, CD ⊥ OA, and the perpendicular foot is D, then the value of cos ⊥ AOB is equal to () A. OD B. OA C. CD D. AB
∵CD⊥OA，
∴∠CDO=90°，
∵OC=1，
∴cos∠AOB=OD：OC=OD．
Therefore, a
The radius OA of circle O is perpendicular to chord BC. Ad = 2cm BC = 8cm. Find the radius of circle o
D point should be foot drop
Let the radius of circle o be x, then connect ob = X
∵ the radius of circle O is perpendicular to chord BC ᙽ BD = CD = 4, OD = oa-od = X-2
∴ （x-2）^2+4^2=x^2
The solution is x = 5
Given the radius OA ⊥ ob, C and D are the two trisection points of arc ab. AB intersects OC and OD with points E and f respectively to prove that AE = BF = CD
Let's do CD through O. the vertical line crosses AB to G and CD to H
∠COH=∠DOH
Ho vertical ab
AOE ≌BOF
AE=BF
In the circle O, the radius OA is perpendicular to ob, C and D are the trisection points of arc AB, and ab intersects OC at points E and f respectively
Because D, C are the points of the arc AB, so the angle BOD = 30 degrees, 0A = OB and OA is perpendicular to ob, so the angle oba = 45 degrees, so the angle BFD = 30 + 45 = 75 degrees. Because the angle BDO = (180-angle hog) / 2 = (180-30) / 2 = 75 degrees, so the angle BDF = angle BFD, so BD = BF. Similarly, it can be proved that AE = AC. because C, D are the points of arc AB, so arc BD = arc DC = arc Ca, so BD = BD = arc DC = arc Ca, so BD = arc BD = arc DC = arc Ca, so BD = AC. because C, D are the three equipoints of arc AB, so arc BD = arc DC = arc Ca, so BD = arc DC = arc Ca, so BD = arc DC = arc Ca, so BD = ac.= CD = CD, And AE = BF = CD
Known: as shown in the figure, AC and BD intersect at point O, AB / / CD, OA = ob, and OC = OD
∵ AB / / CD, so ᙽ OAB ᙽ OCD, oba ∵ ODC, ᙽ OA = ob, OAB = ∠ oba
∴∠OCD＝∠ODC,∴OC=OD
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It is known that: O is the intersection point of the diagonal of rectangular ABCD, e, F, G, h are the points on OA, ob, OC, OD respectively, AE = BF = CG = DH verification: quadrilateral efgh is a rectangle It is known that: O is the intersection point of the diagonal of rectangular ABCD, e, F, G, h are the points on OA, ob, OC, OD respectively, AE = BF = CG = DH verification: quadrilateral efgh is a rectangle
Connect EF, FG, GH, he
Because ABCD is a rectangle, so Ao = Bo, and because AE = BF, EO = fo. Similarly, EO = fo = go = ho. Therefore, the diagonals of quadrilateral efgh are bisected and equal, so quadrilateral efgh is rectangular
RELATED INFORMATIONS
- 1. It is known that: as shown in the figure, take the oblique edge ab of RT △ ABC as the diameter to make ⊙ o, and D is the point on ⊙ o, and there is AC = CD. Pass through point C to make the tangent of ⊙ o, and connect CD with the extension line of BD at point E (1) Try to judge whether be and CE are perpendicular to each other, please explain the reason; (2) If CD = 2 5，tan∠DCE=1 2. Find the radius length of ⊙ o
- 2. Given that AB and CD are two diameters of circle O, chord CE / / AB, angle COE = 50 degrees, then the acute angle DOB =?
- 3. It is known that, as shown in the following four graphs, circle O1 and circle O2 intersect at two points a and B, the line CD passing through point a intersects with circle O1 at C, intersects with circle O2 at D, and straight line ef? 2 passing through point B As shown in the following figure, circle O1 and circle O2 intersect at two points a and B, the line CD passing through point a intersects with circle O1 at C, and with circle O2 at D. the line EF passing through point B intersects with circle O1 at e and circle 02 at F, so as to verify CE ∥ DF Picture in space photo album
- 4. As shown in the figure, the diameter of ⊙ o is ab = 10, C and D are two points on the circle, and AC＝ CD＝ Let the tangent ed of point d cross the extension line of AC at point F. connect OC to ad at point G (1) Confirmation: DF ⊥ AF (2) Seek the length of OG
- 5. As shown in the figure, it is known that AB is the chord of ⊙ o, the radius OA = 20cm, ∠ AOB = 120 ° and the area of ⊙ AOB is obtained
- 6. In the RT triangle ABC, the angle ABC = 90 degrees, the circle O with the diameter of AB intersects AC with D, and the tangent passing through D intersects BC with e. it is proved that de = 1 / 2 BC, If Tan angle c = root 5 divided by 2, de = 2, find ad
- 7. As shown in the figure, the straight line DF intersects with AB and AC on both sides of △ ABC at two points D and e respectively, and intersects with the extension line of BC at point F, ∠ B = 50 °, 1 = 76 °, f = 3
- 8. As shown in the figure, in △ ABC, D is a point on AC, e is a point on CB extension line, and AC BC＝EF DF， Confirmation: ad = EB
- 9. As shown in the figure, it is known that AB is the diameter of ⊙ o, AC is the chord, and bisection ⊥ bad, ad ⊥ CD, and the perpendicular foot is d Proof: CD is tangent of ⊙ o
- 10. When the radius of a circle is several centimeters, its circumference and area are equal
- 11. As shown in the figure, AB is the diameter of semicircle o, and point C is a point on the semicircle. Through point C, make CD perpendicular to D, AC = 2 times, root sign 3cm, ad: DB = 3:1, and find the length of AD As shown in the figure, AB is the diameter of semicircle o, and point C is a point on the semicircle. Through point C, make CD perpendicular to D, AC = 2 times, root sign 3cm, ad: DB = 3:1, and find the length of AD
- 12. As shown in the figure, ad is the diameter of circumscribed circle of △ ABC, CE ⊥ ad intersects ad with F and ab with E. try to explain that AC square = ab × AE
- 13. As shown in the figure of triangle ABC, ad is the center line on BC side, f is a point on ad, and AF: FD = 1:3, extend BF, cross AC to e, and find AE: EC
- 14. It is known that AB is the diameter of circle O, PD tangent circle O to C, and the extension line of Ba intersects PC at P Angle P = 26 degrees, calculate angle BCD
- 15. As shown in the figure, it is known that ⊙ o is the circumscribed circle of ⊙ ABC, CE is the diameter of ⊥ o, CD ⊥ AB and D are perpendicular feet. It is proved that ⊙ ACD = ∠ BCE
- 16. As shown in the figure, the triangle ABC is inscribed with the circle O, ad bisects ∠ BAC, intersects ⊙ O and D, passes through D as de ∥ BC, and intersects the extension line of AC with E The triangle ABC is inscribed with the circle O, ad bisects ⊙ BAC, intersects ⊙ O and D, passes D as de ∥ BC, and intersects the extension line of AC with E Try to judge the position relationship between de and circle O and prove your conclusion 2. If ∠ e = 60 degrees, the radius of circle O is 4, and find the length of ab
- 17. O is the diameter of a perpendicular point on each other (1) Verification: AC bisection ∠ DAB; (2) If CD = 4, ad = 8, try to find the radius of ⊙ o
- 18. As shown in the figure, the vertical bisector of chord AB intersects at point C and intersects chord AB at point D. It is known that ab = 24cm, CD = 8cm (1) Find the circle where the fragment is located (keep the drawing trace and write the method); (2) Find the radius of the circle in (1)
- 19. As shown in Figure AC is the diameter of circle O, PA is perpendicular to AC, connecting OP, chord CB / / op, diameter BC intersects straight line AC at D, BD = 2PA, prove that BP is tangent of circle O, OP is related to BC Verify the value of sin angle OPA I 1 and can not insert the picture The diameter BC in the problem is changed to the straight line Pb
- 20. As shown in the figure, the radius of ⊙ o is 1cm, and the lengths of strings AB and CD are respectively 2 cm, 1 cm, then the acute angle α between the strings AC and BD=______ Degree