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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
Connect OC
In RT △ OCM, OC = 6, OM = 1
4AB=3，
According to Pythagorean theorem, MC=
OC2−OM2=3
3；
∴CD=2MC=6
3．
Therefore, D
As shown in the figure, in circle O, it is known that AB is the diameter, the chord CD intersects AB at P, and P is the midpoint of ob, so find the value of Tanc × tand This is the picture
When BC and BD are connected, AB is the diameter of the circle
tanC=tan∠ABD=AD/BD
tanD=tan∠ABC=AC/BC
So Tanc * tand
=(AD*AC)/(BD*BC)
=S(△ACD)/S(△BCD)
=AD/BD
Because D is the midpoint of ob, then ad / BD = 3
There is Tanc * tand = 3
AB is the chord of circle O, P is a point on the chord ab of circle O, AB 10cm, AP 4cm, Op 5cm, then the radius of circle O is () cm ditto,
First of all, we should know that the vertical line from the center of the circle to the chord is the vertical bisector of the string. Then, if we make the vertical line of the string through the center of the circle, we can get a right triangle composed of OP and vertical line. OP = 5, bottom = 5-4 = 1, then the perpendicular can be calculated by the Bishop's theorem. Since the vertical line is out, the radius of the large triangle composed of radius and vertical line can be calculated by the same method
As shown in the figure, the diameter ab of the center O is ab = 16, P is the midpoint of ob, and the angle between the chord CD passing through point P and ab is APC = 30 degrees, and the length of CD can be calculated If the isosceles triangle ABC is inscribed with the center O, ab = AC, the angle BAC = 120 degrees, and ab = 4, then the diameter of the center O is?
In RT △ OPH, ∠ APC = 30 ° OP = 4 AB = 4, then Oh = 2 (half of the slant side). Then connect OD, in RT △ ODH, OD is circular radius, OD = 8, oh = 2, according to Pythagorean theorem
DH = √ 60, similarly to OC, ch = √ 60, so CD = 4 √ 15
As shown in the figure, if the diameter of ⊙ o is 50, point P is the midpoint of chord AB, and chord CD passes through point P, and ab = 40, CD = 48, then cos angle APC = () process
Let the midpoint of CD be q, Op = √ (OA ^ 2-op ^ 2) = √ (25 ^ 2-20 ^ 2) = 15, OQ = √ (OC ^ 2-cq ^ 2) = √ (25 ^ 2-24 ^ 2) = 7. Cos ∠ APC = cos (90 - ∠ OPD) = sin ∠ OPD = OQ / op = 7 / 15
As shown in the figure, in circle O, AB is the diameter, P is the midpoint of ob, ab = 8, the chord CD intersects AB at P, ∠ APC = 30 degrees, find CD
O for OE ⊥ CD, CD for E
∵ diameter AB = 8
∴OB＝4
∵ P is the midpoint of ob
∴OP＝OB/2＝4/2＝2
∵∠APC=30,OE⊥CD
∴OE＝OP×sin30=2×1/2=1
∴CE²=OC²-OE²=16-1=15
∴CE=√15
∵OE⊥CD
∴CD=2CE=2√15
The two chords AB and CD of circle O are perpendicular to point P, AP = 4, BP = 6, CP = 3, DP = 8. Find the radius of circle o
Make om perpendicular to AB and m, on to CD and n
AB and CD are the two chords of circle O, ab = 10, CD = 11
According to the vertical diameter theorem
AM=MB=5,CN=ND=5.5
∵ AB, CD vertical
The ompn is easy to see as rectangular
PM = on = 5-4 = 1
If OC is connected, OC is the radius
OC^2=ON^2 +CN^2
It is calculated by substitution
OC = (5 √ 5) / 2, radius is (5 √ 5) / 2
As shown in the figure, the two chords AB and CD of ⊙ o are perpendicular to point P, AP = 4, BP = 6, CP = 3, DP = 8. Find the radius of ⊙ o If you can't paste the picture, please forgive me
A more troublesome way
Finding angle BDP and CDA by inverse trigonometric function
Pass a to make the vertical line intersect with E
AEB=ADB
Using trigonometric function to find diameter
As shown in the figure, in the two concentric circles with point o as the center, the chord ab of the large circle is the tangent line of the small circle, and the point P is the tangent point
Proof: connect OP as shown in the figure,
∵ the chord ab of the big circle is the tangent line of the small circle, and the point P is the tangent point,
∴OP⊥AB，
∵ OP over O,
∴AP=BP．
As shown in the figure, in the two concentric circles with point o as the center, the chord ab of the large circle is the tangent line of the small circle, and the point P is the tangent point
Proof: connect OP as shown in the figure,
∵ the chord ab of the big circle is the tangent line of the small circle, and the point P is the tangent point,
∴OP⊥AB，
∵ OP over O,
∴AP=BP．
RELATED INFORMATIONS
- 1. 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
- 2. 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
- 3. As shown in the figure, AB and CD are the two chords of circle o
- 4. 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?
- 5. 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．
- 6. 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
- 7. 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
- 8. 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
- 9. 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．
- 10. As shown in the figure AB is the diameter of circle O, AC, AD are chords, and ab bisection angle CAD
- 11. 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
- 12. 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
- 13. 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
- 14. 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!
- 15. 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?
- 16. 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
- 17. 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
- 18. 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
- 19. 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
- 20. In the triangle ABC, ab = 15, AC = 13, height ad = 12, find the circumference of triangle ABC