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As shown in FIG. 11, AB is the diameter of circle O, passing through point o as parallel line of chord BC, tangent AP crossing point a, connecting point P and connecting AC. (1) verification: △ ABC ~ △ Poa (2) OB = 2, Op = 7 / 2, find the length of BC
（1）
prove:
∵ AB is the diameter of circle o
∴∠ACB=90º
∵ AP is the tangent of circle o
∴∠PAO=90º=∠ACB
∵BC//OP
∴∠ABC=∠POA
∴⊿ABC∽⊿POA（AA‘）
（2）
∵OB=2
∴AB=4,OA=2
∵⊿ABC∽⊿POA
∴BC/OA=AB/OP
∴BC=OA×AB/OP
=2×4÷(7/2)=16/7
It is known that the chord ab of circle O is perpendicular to the diameter CD, the perpendicular foot is f, and the point E is on AB, and EA = EC. It is proved that the square of AC = AE times ab 2: Extend EC to point P and connect Pb. If Pb = PE, judge the position relationship between Pb and circle O, and explain the reason
Because the chord AB is vertical diameter CD, so AC = CB, so the angle cab = angle CBA because EA = EC, so the angle EAC = angle ace, so the isosceles triangle ace is similar to the isosceles triangle type ABC, so AC: EC = AB: AC = AC = EC * AB because EA = EC, AC = EA * AB, so the angle cob = 2 times angle CAE because of isosceles triangle type
It is known that the chord ab of circle O is perpendicular to the diameter CD, the perpendicular foot is f, and the point E is on AB, and EA = EC. Extend EC to point P and connect PB to make Pb = PE. Is ac2 = AE * AB
It is known that EA = EC, we can get: ∠ ace = ∠ CAE
If CD is the vertical bisector of AB, AC = BC, then: ∠ BAC = ∠ ABC
In △ ace and △ ABC, ∠ ace = ∠ CAE = ∠ BAC = ∠ ABC,
So, △ ace ∽ ABC,
AC / AB = AE / AC;
That is: AC ^ 2 = AE × ab
The diameter of ⊙ o middle chord ab ⊥ CD is known. The perpendicular foot is point F and the point E is on ab. EA = EC. Verification: AC * AC = AE * ab Extend EC to P and connect Pb. Let Pb = PE. Judging the position relationship between Pb and circle
Because of the chord ab ⊥ diameter CD, so CD is vertically bisected AB, AC = BC ﹣ a = ∠ B, and because EA = EC, so ∠ a = ∠ ace, so, △ ABC ~ △ ace, AC / AB = AE / ACAC * AC = AE * ab
The chord AB is perpendicular to the diameter of circle O CD on F, E on AB and EA = EC. Extend EC to point P so that Pb = PE. Is Pb tangent to circle O?
Answer: Pb is the tangent connection of the circle O, ob, ∵ EC = EA, ? EC = EA, ? EAC, ? EAC, ? EAC, ? EAC, ? EAC, ? EAC, ? Pb = 2 ? EAC, ? EC = EA, ? EAC, ? EAC, 57575757575757575757eeec = EA, EEEC = EA, ? EAC,. Ob ⊥
In circle O, AB is the diameter of circle O, CD is the chord, points E and F are on AB, EC is vertical CD FD vertical CD Confirmation: AE = BF
You just draw a diameter across the center of the CD, G, and o
Then make the vertical lines EH and FL from point E and F to this diameter
We can get rectangular cghe and rectangular gdfl, which leads to eh = fl
The triangle EHO and Flo are similar triangles, and eh = fl
So OE = of, then AE = BF
As shown in the figure, the diameter ab of ⊙ o is perpendicular to the chord CD, the perpendicular foot P is the midpoint of ob, CD = 6cm, and find the length of diameter ab
Connect OC, as shown in the figure,
∵ AB is perpendicular to the chord CD,
∴PC=PD，
CD = 6 cm,
∴PC=3cm，
And ∵ P is the midpoint of ob,
∴OB=2OP，
∴OC=2OP，
∴∠C=30°，
∴PC=
3 OP, then op=
3cm，
∴OC=2OP=2
3cm，
So the length of diameter AB is 4
3cm．
Given that the diameter of circle O is 50cm, the two parallel chords of circle O, ab = 40cm and CD = 48CM, calculate the distance between chord AB and string CD
According to the deduction of the vertical diameter theorem, AE = be = 1 / 2 ab = 20cm CF = DF = 1 / 2 CD = 24cm then OE = 15cm in RT △ OFC = 7cm (1) AB, Cd on the same side of the circle center
As shown in the figure, AB is the diameter of circle O, AB is perpendicular to chord CD at point P, and P is the midpoint of radius ob, CD = 6cm, then the diameter of circle O is
2*3^0.5
As shown in the figure, AB is the diameter of ⊙ and the chord CD is vertically bisected ob, then ∠ BDC = () A. 15° B. 20° C. 30° D. 45°
Connect OC, BC
∵ chord CD vertical bisection ob
∴OC=BC
∵OC=OB
The △ OCB is an equilateral triangle
∴∠COB=60°
∴∠D=30°．
Therefore, C
RELATED INFORMATIONS
- 1. As shown in the figure, in ⊙ o with diameter AB = 12, chord CD ⊥ AB is at m, and M is the midpoint of radius ob, then the length of chord CD is () A. 3 B. 3 Three C. 6 D. 6 Three
- 2. If point P is the point on the chord AB, connect OP, pass P as PC ⊥ OP, and PC intersect ⊙ o at point C. If AP = 4, Pb = 2, then the length of PC is () A. Two B. 2 C. 2 Two D. 3
- 3. The edge ab of the square ABCD with side length of 4 is the diameter of circle O, CF is the tangent line of circle O, e is the tangent point, f is the chord of circle O on AD 1 find the area of CDF 2 find the length of line segment be
- 4. As shown in the figure, AB and CD are the two chords of circle o
- 5. Given that the radius of circle O is 6 and the length of chord AB is 6 times the root sign 3, then the degree of the center angle of chord AB is?
- 6. If the radius of the circle is 2cm, the length of one chord in the circle is 2 3 cm, then the distance between the midpoint of the chord and the midpoint of the inferior arc is______ cm．
- 7. As shown in the figure, circle O1 and circle O2 are inscribed at point a, O2 is then circled O1, and chord AB intersects circle O1 at C. If AB = 10cm, find the length of BC
- 8. For example: the radii of circle O1 and circle O2 are 4 and 1 respectively, O1O2 = 6, P is the moving point on circle O2, and the tangent line of circle O1 is made through point P, then the shortest tangent length is? Picture It should be able to draw it
- 9. As shown in the triangle ABC, ab = 4, the circle O1 with BC as the diameter intersects AC, and D is the midpoint of AC, and de ⊥ ab. if BC ⊥ AB, circle O2 is circumscribed with circle O1, If BC ⊥ AB, circle O2 is circumscribed with circle O1, and tangent to de and be, the radius of circle O2 is calculated
- 10. Known: as shown in the figure, ⊙ O1 and ⊙ O2 intersect at a and B, point O2 is on ⊙ O1, ad is the diameter of ⊙ O2, connect dB and extend ⊙ O1 to point C Verification: 1 2AD＝ CD2−CO22．
- 11. At ⊙ o, C is the center of ACB, CD is the diameter, chord AB intersects CD at point P, PE is perpendicular to point E, if BC = 10, CE ratio EB = 3:2, AB is found
- 12. As shown in the figure, in the triangle ABC, ab = AC, BF = CD, BD = CE, given the angle a = 40, calculate the degree of the angle EDF
- 13. In the triangle ABC, ad is the center line on BC side, e is a point on AC, be and ad intersect F, if AE = EF, it is proved that BF = AC Here I know to extend ad, but I don't know if you can make CG through point C, hand over ad to g, make CG = DC, I don't know if this is OK. Ask!
- 14. What is the relationship between the degree of angle AEC and the degree of arc AC and arc BD? If e is outside the circle, what is the relationship between the degree of angle AEC and the degree of arc AC and arc BD?
- 15. As shown in FIG. 12, the diameter of circle O1 AB and chord CD are compared with point E. we know that AE = 6cm, EB = 2cm, ∠ CEA = 30 degrees. Find the length of CD
- 16. As shown in the figure, AB is a chord of ⊙ o, OD ⊥ AB, perpendicular foot is C, intersect ⊙ o at point D and point E on ⊙ o (1) If ∠ AOD = 52 °, calculate the degree of ∠ DEB; (2) If OC = 3, OA = 5, find the length of ab
- 17. As shown in the figure, △ ABC, e is the heart of △ ABC, the bisector of ∠ A and the circumscribed circle of △ ABC intersect at point D. It is proved that de = dB
- 18. As shown in the figure, in the acute triangle ABC, AB > AC, ad is perpendicular to D, and graph o with AD as diameter intersects AB, AC with E and f respectively
- 19. In the triangle ABC, ab = 15, AC = 13, height ad = 12, find the circumference of triangle ABC
- 20. Take the right angle side of the RT triangle ABC as the diameter, make a semicircle o, intersect the oblique side with D, OE parallel with AC, and intersect AB with e. it is proved that De is the tangent of circle o