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As shown in the figure, in ▱ ABCD, AE bisects ∠ bad to intersect DC at point E, ad = 5cm, ab = 8cm, and calculate the length of EC
In the parallelogram ABCD, then ab ∥ CD,
∴∠2=∠3，
AE bisects ∠ bad, i.e. ∠ 1 = ∠ 3,
Ψ 1 = ∠ 2, that is, de = ad,
Ad = 5cm, ab = 8cm,
∴EC=CD-DE=8-5=3cm．
Therefore, the length of EC is 3cm
As shown in the figure, AE and AF are the heights on BC and CD sides respectively, AE = 2cm, AF = 5cm, EAF = 30 ° to find the inner angles of each parallelogram ABCD
In the quadrilateral aecf, (it is estimated that the midpoint e of the graph is on BC, and the point F is on CD), we know that ∠ EAF = 30 ° and ∠ AEC = ∠ AFC = 90 °; it can be concluded that: ∠ C = 360 ° - ∠ EAF - ∠ AEC - ∠ AFC = 150 °; therefore, the sizes of the inner angles of the parallelogram ABCD are: ∠ a = ∠ C = 150 °, ∠ B = ∠ d =
As shown in the figure, in the rectangle ABCD, ab = 3, ad = 2, points E and F are on AB and DC respectively, AE = DF = 2. Now, place a protractor with diameter of 2 (center of circle is o) on the figure, so that its 0 ° line Mn coincides with EF; if the endpoint n on the 0 ° line of the protractor is fixed on point F, and then the protractor is rotated clockwise about point F by ∠ α (0 ° < α < 90 °), at this time, the semicircle of the protractor intersects with EF Point P, let the protractor read n ° at point P (1) An algebraic expression containing n ° is used to express the size of ∠α; (2) When n ° is equal to, the segment PC is parallel to MF? (3) In the course of the rotation of the protractor, go through the point m 'as GH ⊥ m ′ F, intersect AE at point G and cross ad at point h. let the area of Ge = x and △ AGH be s, try to find out the function relation of s with respect to x, and write out the value range of independent variable x
(1) Connect o ′ P, then ∠ Po ′ f = n °;
∵O′P=O′F，
∴∠O′FP=∠a，
ν n ° + 2 ∠ α = 180 °, i.e. ∠ α = 90 ° - 1
2n°；
(2) Connect m'p, PC
∵ m ′ f is the diameter of the semicircle o ′,
∴M′P⊥PF；
And ∵ FC ⊥ PF,
∴FC∥M′P，
If PC ‖ m ′ F,
The quadrilateral m ′ PCF is a parallelogram, ∠ α = 30 °,
∴PC=M′F=2FC，∠α=∠CPF=30°；
Substituting into the relation in (1), we can get:
30°=90°-1
2n°，
That is, n ° = 120 °;
(3) Draw an arc ed with point F as the center of the circle and the length of Fe as the radius;
∵ GM ⊥ m ′ f at point m ′,
/ / GH is the tangent of the arc ed,
Similarly, GE and HD are tangent lines of arc ed,
∴GE=GM′，HM′=HD；
Let Ge = x, then Ag = 2-x,
Let DH = y, then HM '= y, ah = 2-y;
In RT △ AGH, ag2 + ah2 = GH2
（2-x）2+（2-y）2=（x+y）2
That is: 4-4x + x2 + 4-4y + y2 = x2 + 2XY + Y2
∴y=4−2x
X+2
∴S=1
2AG•AH=1
2（2-x）（2-y）=4x−2x2
x+2（0＜x＜2）
That is, the function relationship between S and X is s = 4x − 2x2
x+2（0＜x＜2）．
Known: as shown in the figure, the quadrilateral ABCD is a rectangle (AD > AB), the point E is on BC, and AE = ad, DF ⊥ AE, the perpendicular foot is F. please explore the quantitative relationship between DF and ab? Write down the conclusions you have come to and prove them
The conclusion is: DF = ab. (1 point)
Prove: the quadrilateral ABCD is a rectangle
∴∠B=90°，AD∥BC．
﹤ DAF = ∠ AEB. (2 points)
∵DF⊥AE．
∴∠AFD=90°．
∵AE=AD．
≌△ DFA. (5 points)
ν AB = DF. (6 points)
As shown in the figure, the edge ab of rectangular ABCD passes through the center of ⊙ o, e and F are the intersection points of AB, CD and ⊙ o respectively. If AE = 3cm, ad = 4cm, DF = 5cm, then the diameter of ⊙ o is equal to______ .
Connect of, as FG ⊥ AB at point G
Then eg = df-ae = 5-3 = 2cm
Let the radius of ⊙ o be r,
Then of = R, og = R-2
In the right angle △ OFG, of2 = fg2 + og2,
R2 = (R-2) 2 + 42,
The solution is: R=5
The diameter is 10 cm
So the answer is: 10
As shown in the figure, the edge ab of rectangular ABCD passes through the center of ⊙ o, e and F are the intersection points of AB, CD and ⊙ o respectively. If AE = 3cm, ad = 4cm, DF = 5cm, then the diameter of ⊙ o is equal to______ .
Connect of, as FG ⊥ AB at point G
Then eg = df-ae = 5-3 = 2cm
Let the radius of ⊙ o be r,
Then of = R, og = R-2
In the right angle △ OFG, of2 = fg2 + og2,
R2 = (R-2) 2 + 42,
The solution is: R=5
The diameter is 10 cm
So the answer is: 10
In a circle with radius 5cm, chord AB = 6, chord CD = 8cm, and ab is parallel to CD. Find the distance between AB and CD?
7cm,1cm
If ⊙ o radius is 5cm, chord AB / / CD, and ab = 8cm, CD = 6cm, then the distance between AB and CD is__________ Cm. this is it
In this paper, we should pay attention to two cases: first, when AB and CD are on both sides of the center of a circle, we make OE ⊥ AB through O, the foot of foot is e, intersect CD with point F, because AB / / CD, OE ⊥ AB, so of ⊥ CD is obtained from the vertical diameter theorem: AE = 1 / 2, ab = 4 in RT △ OAE, OE = 3 by Pythagorean theorem, and in RT △ OCF, of = 4
If the lengths of two parallel chords are 6cm and 8cm respectively in a circle with a radius of 5cm, the distance between the two chords is______ .
The distance from the center of the circle to the two chords is D1=
52−(1
2×6)2=4cm，d2=
52−(1
2×8)2=3cm．
Therefore, the distance between two strings d = d1-d2 = 1cm or D = D1 + D2 = 7cm
There are two parallel chords in circle O with a radius of 5cm, and the lengths are 8cm and 6cm respectively. The distance between these two chords is a complete process. Thank you
There are two cases
When two parallel chords are on the same side of the center of the circle,
The distance between the two chords = √ (5? - 3?) - √ (5? 2 - 4?) = 4-3 = 1cm
When two parallel chords are not on the same side of the center of the circle,
The distance between the two chords = √ (5? - 3?)) + √ (5? - 4?) = 4 + 3 = 7cm
RELATED INFORMATIONS
- 1. If the lengths of two parallel chords are 6cm and 8cm respectively in a circle with a radius of 5cm, the distance between the two chords is______ .
- 2. It is known that, as shown in the figure, the diameter ab of the circle O intersects with the chord CD at the point E, AE = 1, be = 5, ∠ AEC = 30 ° and find the length of CD I can't draw a picture
- 3. 0
- 4. If AB is 10, Pb is 4 and OP is 5, then find the radius of the circle
- 5. It is known that in the triangle ABC of RT, the angle ACB = 90 degrees, D and E are the midpoint of AB and BC respectively, and the point F is on the extension line of AC, and CF = De
- 6. Find a proof question: AB is the diameter of circle O, CD is a chord in the circle, AE ⊥ CD is in E, BF ⊥ CD is in F, and CE = DF
- 7. In ⊙ o, the chord ab ⊥ CD is in M. the radius of ⊙ o is 5, BM = 6, am = 2. Find the size of dm-cm
- 8. As shown in the figure, in circle O, AB is the diameter of circle O, CD is a chord, and CD is perpendicular to point P, connecting BC and AD
- 9. Enumerate some algorithms under radical
- 10. If the perimeter of a right triangle is 4 + 2, the root is 6, and the length of the central line on the hypotenuse is 2, then the area of the triangle is
- 11. In circle O, the chord Mn / / EF, P is a point on circle O. the radius of circle O is known to be 10cm, Mn = 12cm, EP = 16cm. Find the distance between chord Mn and ef
- 12. As shown in the figure, in the plane rectangular coordinate system, the two sides of the rectangular oabc are respectively on the x-axis and y-axis, OA = 10 cm, OC = 6 cm, and the existing two moving points P and Q are from the As shown in the figure, in the plane rectangular coordinate system, the two sides of the rectangular oabc are respectively on the x-axis and y-axis, OA = 10 cm, OC = 6 cm. The existing two moving points P and Q start from D and a at the same time, point P moves uniformly along the direction of OA on line OA, and point Q moves uniformly along AB direction on line ab. the velocity of point P is known to be 1cm / s (1) Suppose that the velocity of point q is cm / s and the movement time is T seconds, ① When the area of △ CPQ is the smallest, find the coordinates of point Q; ② when △ cop and △ PAQ are similar, find the coordinates of point Q (2) Suppose the velocity of point q is a cm / s, ask if there is a value of a, so that △ OCP is similar to △ PAQ and △ CBQ? If so, ask for the value of a and write out the coordinates of point Q at this time; if not, please explain the reason
- 13. Given that ∠ AOB = 90 degrees, D, C divide AB into three equal parts, chord AB intersects with radius OC, OD at points E and F, and proves that AE = DC = BF
- 14. As shown in the figure, OA ⊥ OC, ob ⊥ OD, ∠ BOC = 30 ° and find AOB, COD, AOC
- 15. In the circle O, AB is the diameter, the chord CD is perpendicular to P, AC = DC = 2 times the root sign 3, and find the length of Op
- 16. As shown in the figure, ad is the diameter of ⊙ o, AC is the chord, ∠ CAD = 30 °, ob ⊥ ad is in O, AC is in B, ab = 5, and the length of BC is obtained
- 17. Circle O1 and circle O2 intersect at two points a and B, P is the point on circle O1, connecting PA and Pb, intersecting circle O2 with C and D. It is proved that po1 is perpendicular to CD
- 18. As shown in the figure, point h is the vertical center of △ ABC, the circumscribed circle ⊙ O2 of ⊙ O1 and ⊙ BCH with ab as diameter intersect at point D, and extend ad to intersect ch at point P, Verification: point P is the midpoint of Ch
- 19. As shown in the figure, circle I is the inscribed circle of the triangle ABC, which is tangent to points D, e and F with AB, BC and Ca respectively, and the angle def = 50 degrees. Find the angle A
- 20. It is known that the radii of circle O1 and circle O2 are 3cm and 5cm respectively, and if they are inscribed, the position relationship between the center of circle and O1O2? What is the positional relation? Which is the extrinsic intersection and tangent? It is known that the radius of circle O1 and circle O2 are 3cm and 5cm respectively, and if they are inscribed, the center distance of circle is 5cm!