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It is known that the length of the two chords of a circle AB CD is two of the equation x ^ 2-42x + 432 = 0, and ab is parallel to CD, and the distance between the two chords is 3. Find the radius
Solve the equation first, and get three solutions X1 = 18, X2 = 24
If AB and CD are on the same side of the center of the circle, make the vertical lines AB and CD through the center O, and the vertical feet are e and f respectively
Then oe2 + r = 2 （1）
Assuming that of > OE, that is, AB is far from the center of the circle, replace AF = 24 / 2 = 12, CE = 18 / 2 = 9, of = OE + 3, and find out if it meets the conditions
I have a general solution, which seems to be inconsistent, because of = R, that is, AB is tangent
If AB and CD are on both sides of the circle center, of = 3-oe
By substituting the condition into the solution of of, the radius R is obtained
In addition, similar triangles can be used to solve of, thus the radius can be calculated
Because of the "speed, speed, speed, speed", the written calculation is done by you. I have given you some ideas in the most concise words. Don't be too lazy. Just steal a little
Given that the two chords of a circle AB, CD are two of the equations x2-42x + 432 = 0, and ab ∥ CD, and the distance between the two chords is 3, then the radius length of the circle is
If AB CD is in a semicircle, because ab ‖ CD, then the root (r? - 12) = root (r? - 9? - 3) square on both sides r? - 144
Two known far strings AB.CD The length of is equation x * 2_ If we know that the distance between the two chords is 3, then the radius length of the circle is
X ^ 2-42x + 432 = 0 (x-18) (x-24) = 0, x = 18 or x = 24. If the radius is R and the distance from the center of the circle to the nearest chord is D, then R ^ 2 = D ^ 2 + 12 ^ 2, R ^ 2 = (D + 3) ^ 2 + 9 ^ 2, d = 9, r = 15
The radius of the circle is known to be 5, AB.CD Are two chords of circle O, and ab is perpendicular to CD and E, if AE.BE There are two roots of {{0 +} and {2 +}, Find the chord center distance of CD and the value of K Hurry!
AE.BE Are the two roots of the equation {x} ^ {2} - 8x + {K} ^ {2} = 0, so X1 + x2 = 8, that is ab = 8
So the chord center distance of AB = under the root sign (5 2 - (8 / 2)) = 3
Therefore, the chord center distance of CD = under the root sign (4? - 3?) = root 7
X1 x x2 = k? = ((8 / 2) + radical 7) x ((8 / 2) - radical 7) = 9
So k = 3 or - 3
8. The length of chord AB is equal to? (specific operation process)
The distance from the center of a circle (0,0) to the straight line x + √ 3y-2 = 0 is | 0 + √ 3 * 0-2 / √ (1 + 3) = 1
From the vertical diameter theorem | ab | = 2 √ (2 ^ 2-1 ^ 2) = 2 √ 3
So AB = 2 root sign 3
Hope to adopt, thank you=
As shown in the figure, AB is the chord of ⊙ o, and the radius OC and OD intersect AB at points E and f respectively, and AE = BF. Please find out the quantitative relationship between line OE and of and give proof
OE=OF，
Proof: connect OA, ob,
∵OA=OB，
Ψ OAB = ∠ oba, that is ∠ OAE = ∠ oBf
In △ OAE and △ oBf,
OA=OB
∠OAE=∠OBF
AE=BF ，
∴△OAE≌△OBF（SAS）．
∴OE=OF．
As shown in the figure, in ⊙ o, chord AB and radius OC intersect at point m, and OM = MC. If am = 1.5 and BM = 4, then the length of OC is () A. 2 Six B. Six C. 2 Three D. 2 Two
As shown in the figure, extend CO and intersect ⊙ o to D, then CD is the diameter of ⊙ o;
∵OM=MC，
∴OC=2MC=2OM，DM=3OM=3MC；
According to the intersecting string theorem, DM? MC = am? BM,
That is, 3mc2 = 1.5 × 4=
2；
∴OC=2MC=2
2, so D
If AB is the diameter of circle O, the chord CD ⊥ AB is m, am = 6, BM = 4
∵ AB is the diameter, ab ⊥ CD
∴CM²=AM*BM=6*4=24
ν cm = 2 pieces, 6 pieces
ν CD = 4 root numbers 6
Given the diameter of circle O AB = 10, chord CD ⊥ AB at point m, if om: OA = 3:5, what is the length of AC?
If M is in the middle of OA, OM = 3 / 5oa = 3, MC = root (OC ^ 2 - om ^ 2) = 4am = OA - om = 2, so AC = root (am ^ 2 + cm ^ 2) = 2 root 52. If M is in the middle of ob, am = OA + om = 8ac = root (am ^ 2 + cm ^ 2) = 4 root 5
As shown in the figure, in circle O, CD is the diameter, AB is the chord ab ⊥ CD in M, OM = 3, DM = 2, find the length of chord ab
Om squared + am squared = OA squared
Am square = 5 * 5-3 * 3 = 16
AM=4
AB=AM*2=4*2=8
The length of string AB is equal to 8
RELATED INFORMATIONS
- 1. It is known that the circle centered on point P passes through points a (- 1,0) and B (3,4), and the vertical bisector of line AB intersects the circle P at points c and D, and | CD | = 4 10． (1) Find the equation of straight line CD; (2) Find the equation of circle P; (3) Set the point Q on the circle P. how many points Q have the area of △ qAB equal to 8? Prove your conclusion
- 2. 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
- 3. 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
- 4. 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)
- 5. 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
- 6. 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
- 7. 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
- 8. 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
- 9. 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
- 10. 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
- 11. 2 ax by + 2 = 0 (a > 0, b > 0). The chord length cut by the circle x ^ 2 + y ^ 2 + 2x + - 4Y + 1 = 0 is 4
- 12. What is the chord length of a line passing through the origin and with an inclination angle of 60 degrees cut by the circle x square + y square - 4Y = o?
- 13. Let the midpoint of chord ab of circle x2 + y2-4x-5 = 0 be p (3,1), then the equation of straight line AB is () A. x+y-4=0 B. x+y-5=0 C. x-y+4=0 D. x-y+5=0
- 14. Given that the circle C: x 2 + y 2 = 5, the straight line L: ax-y-2a = 0 and circle C intersect at two points AB, the trajectory equation of the midpoint of chord AB is obtained
- 15. It is known that the circle C passes through point 10 and the center of the circle is on the positive half axis of the X axis. The straight line x + Y-1 = 0 is cut by the circle C, and the chord length is 2, and the circle standard is found equation
- 16. If x ^ 2 + y ^ 2-2y = 0, the midpoint of chord AB is (- 1 / 2,3 / 2), then ab=
- 17. Find the equation of the circle whose center is on the line 3x-y = 0, tangent to the x-axis, and cut by the line X-Y = 0
- 18. The straight line kx-y + 1 = 0 and the circle x? 2 + y? 2 = 1 intersect a and B, and find the locus of the midpoint of chord ab
- 19. If the chord ab of the circle x ^ 2, y ^ 2 = 20 is made through the point P (2, - 3), and P bisects AB, then the equation of the line where the chord AB is located is
- 20. Find a point in the ellipse C: x ^ 2 + 4Y ^ 2 = 16. Find the equation of the line where the chord is located with a point a (1, - 1) in the ellipse x * 2 + 4Y * 2 = 16 as the midpoint