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As shown in the figure, the ray OA, ob, OC, OD have common endpoint o, and ∠ AOB = 90 °, COD = 90 °, AOD = 5 4 ﹤ AOC. calculate the degree of ﹤ BOC
Let ∠ AOC = x °, then ∠ AOD = 5
4x°．
∵∠AOC+∠AOD+∠COD=360°，
∴x+5
4x+90=360，
The solution is x = 120,
∴∠AOC=120°，
∴∠BOC=360°-∠AOC-∠AOB=360°-120°-90°=150°．
As shown in the figure, there are five rays OA, ob, OC, OD, OE with common endpoints in the plane For example, for the five rays OA, ob, OC, OD, OE with common endpoint in the drawing plane, draw a circle with o as the center of the circle, mark the numbers 1,2,3,4,5 at the intersection of the first circle and the rays OA, ob, OC, OD, OE; mark the numbers 6,7,8,9 at the intersection of the second circle and the rays OA, ob, OC, OD, OE, 10. And so on. [1] "13 is at the intersection point of the ray and the first circle; [2] expressed by the formula containing N: the number arrangement rule on ray OA is; the number arrangement rule on ray OE is; [3] guess" 2013 "at the intersection point of which ray and which circle, try to explain the reason
0
From the end point O of the ray OA, two more rays OB and OC are introduced, so that ∠ AOB = 60 ° and ∠ BOC = 15 ° can be drawn according to the meaning of the title, and the value of ∠ AOC can be calculated
There are two cases, one is OC in the middle of ray OA and ray ob, this case is 60-15 = 45, and ∠ AOC is 45 degrees
In the case of AOC = 75, the case of AOC = 75 ° is the case
As shown in the figure, ∠ AOB = 30 ° OC bisection ∠ AOB, P is any point on OC, PD ‖ OA intersects ob in D, PE ⊥ OA in E. if od = 4cm, find the length of PE
Over P as PF ⊥ ob in F, ∵∵ AOB = 30 °, OC bisection AOB, AOC = ∠ BOC = 15 °, PD ∥ OA, DPO = ∠ AOP = 15 °, ∵ BOC = ∠ DPO, ? od = 4cm, ? AOB = 30 °, PD ∥ OA, BDP = 30 °, in RT ⊥ PDF, PF = 12pd = 2cm, OC is an angular bisector
As shown in the figure, ∠ AOB = 30 ° OC bisection ∠ AOB, P is any point on OC, PD ‖ OA intersects ob in D, PE ⊥ OA in E. if od = 4cm, find the length of PE
Over P as PF ⊥ ob in F, ∵∵ AOB = 30 °, OC bisection AOB, AOC = ∠ BOC = 15 °, PD ∥ OA, DPO = ∠ AOP = 15 °, ∵ BOC = ∠ DPO, ? od = 4cm, ? AOB = 30 °, PD ∥ OA, BDP = 30 °, in RT ⊥ PDF, PF = 12pd = 2cm, OC is an angular bisector
As shown in the figure, OA, OB and OC are the radii of circle O, ∠ AOB = 2 ∠ BOC It is proved that: ∠ ACB = 2 ∠ BAC
Proof: ACB = 1
2∠AOB，∠BAC=1
2∠BOC；
And ∵ AOB = 2 ∵ BOC,
∴∠ACB=2∠BAC．
As shown in the figure, OA, OB and OC are the radii of circle O, ∠ AOB = 2 ∠ BOC It is proved that: ∠ ACB = 2 ∠ BAC
Proof: ACB = 1
2∠AOB，∠BAC=1
2∠BOC；
And ∵ AOB = 2 ∵ BOC,
∴∠ACB=2∠BAC．
As shown in the figure, OA, ob, OC are all radius of ⊙ o, ᙽ AOB = 2 ∠ BOC. This paper explores the quantitative relationship between ∠ ACB and ∠ BAC, and explains the reasons
∠ACB=2∠BAC．
Proof: ACB = 1
2∠AOB，∠BAC=1
2∠BOC；
And ∵ AOB = 2 ∵ BOC,
∴∠ACB=2∠BAC．
As shown in the figure, OA, OB and OC are the radii of circle O, ∠ AOB = 2 ∠ BOC It is proved that: ∠ ACB = 2 ∠ BAC
Proof: ACB = 1
2∠AOB，∠BAC=1
2∠BOC；
And ∵ AOB = 2 ∵ BOC,
∴∠ACB=2∠BAC．
As shown in the figure, OA, OB and OC are the radii of circle O, ∠ AOB = 2 ∠ BOC It is proved that: ∠ ACB = 2 ∠ BAC
Proof: ACB = 1
2∠AOB，∠BAC=1
2∠BOC；
And ∵ AOB = 2 ∵ BOC,
∴∠ACB=2∠BAC．
RELATED INFORMATIONS
- 1. As shown in the figure, OA, OB and OC are the radii of circle O. if ∠ AOB = 2 ∠ BOC, please judge whether ∠ ACB = 2 ∠ BAC is tenable. Why?
- 2. The right angle vertex P of the triangle ruler slides on the ray OM, and the two right angle sides intersect OA and ob at point CD respectively Conjecture the quantitative relationship between PC and PD
- 3. As shown in the figure, the side of the cone is cut along OA, and the fan-shaped OAB is expanded into a plane figure (1) What is the relationship between the length of the sector arc AB and the length of the circle at the bottom of the cone? What is the position relationship between point a and point B on the side of the cone? (2) If the angle ∠ AOB = 90 °, what is the relationship between the radius r of the cone bottom circle and the radius r of fan-shaped OAB (i.e. OA or OB)? (3) If point a moves on the side of the cone and then returns to its original position, how should the shortest distance of point a be designed? If R 2 = 0.5, ∠ AOB = 90 °, find the shortest distance of point a
- 4. As shown in the figure, it is known that ∠ AOB = 30 ° and the point P is inside ∠ AOB, Op = 6. If there is a moving point m on OA and a moving point n on ob, then the minimum perimeter of △ PMN is______ .
- 5. In the triangle ABC, if AC = radical 2, BC = radical 7, ab = 3, then cosa =?
- 6. Known: as shown in the figure, in the right triangle ABC, the angle c = 90 degrees, the point O is the center of the circle, the length of OA is the radius of the circle, and AC, AB intersect with point D.E, and the angle CBD = angle a.bd is the tangent line of circle O. find: if ad: Ao = 8:5, BC = 2, find the length of BD
- 7. Point O is a point in △ ABC, ∠ ABO = 25 degrees, ∠ ACO = 35 degrees, and the degree of ∠ BOC is 5 degrees less than 2 times of ∠ a In △ ABC, the top angle is ∠ a, the left is ∠ ABC, and the right is ∠ ACB
- 8. In △ ABC, if the bisectors of ∠ a = 50 °, B and C intersect at point O, then the degree of ∠ BOC is () A. 65° B. 100° C. 115° D. 130°
- 9. As shown in the figure, in the RT triangle ABC, the angle B = 90 degrees. If the distance between the point O and the triangle three times is equal, what is the angle AOC? For detailed explanation
- 10. It is known that: as shown in the figure, in △ ABC, the bisector of ∠ B and the bisector of external angle of △ ABC intersect at point D, ∠ a = 90 °
- 11. In the plane rectangular coordinate system, the image of function y = - 3 / 4x + 6 intersects X axis and Y axis at point a and B respectively, line BC and X axis intersect at C, and point C is the midpoint of line OA Find out whether there is a point m on the straight line AB so that △ BCM is an isosceles triangle. If there is, find out the coordinates of point M
- 12. Known: as shown in the figure, find a point P on the straight line Mn, so that the distance between the point P and the two sides of ∠ AOB is equal (the method is required to be written, the drawing trace shall be retained, and the conclusion shall be written)
- 13. As shown in the figure, ∠ AOB = 30 ° OC bisection ∠ AOB, P is a point on OC, PD ‖ OA intersects ob at point D, PE ⊥ OA at e, OD = 4cm, then PE ⊥ OA is at point E=______ .
- 14. The bottom surface of four pyramids P-ABCD is square, and the bottom surface of PB ⊥ ABCD proves that no matter how the height of four pyramids PB changes, the surface PAD and the surface PCD cannot be vertical
- 15. It is known that: as shown in the figure, P is the interior point of the square ABCD, ∠ pad = ∠ PDA = 15 °. Verification: △ PBC is an equilateral triangle
- 16. Ray OA, ob, OC, OD have a common endpoint o, and OA is perpendicular to ob, OC is perpendicular to OD, ∠ AOD = 5 / 4 ∠ AOC, calculate the degree of ∠ BOC ∠ AOD = 5 / 4 ∠ AOC! No, ∠ AOD = 5 / 4 ∠ BOC
- 17. As shown in Fig
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- 19. Mathematical problem: introduce two tangent lines PA, Pb from the moving point P to the circle x ^ 2 + y ^ 2 = 1 1, introduce two tangent PA, Pb from the moving point P to the circle x ^ 2 + y ^ 2 = 1, and the tangent points are a, B, and ∠ APB = 60 degrees, then the trajectory equation of the moving point P is________ 2. Let P (x, y) be a moving point on the circle x ^ 2 + (Y-1) ^ 2 = 1. If the inequality x + y + C > 0 holds, then the value range of C_____________ 3. We know that the equation of the line where the edge ab of the equilateral △ ABC is located is x √ 3 + y = 0, and the coordinates of point C are (1, √ 3) 4. It is known that the circle M: x ^ 2 + (Y-2) ^ 2 = 1, q is the moving point on the x-axis, QA and QB tangent the circle m to two points a and B respectively (1) (liberal arts) if | ab | = 4 √ 2 / 3, find the equation of line MQ (Science) prove that the straight line AB always crosses a fixed point (2) Find the locus equation of midpoint P of moving string ab It's better to analyze it
- 20. As shown in the figure, lead the tangent PA, Pb of the circle through a point p outside ⊙ o with radius of 6cm, connect Po, intersect ⊙ o at F, make tangent of ⊙ o through F, intersect PA and Pb at D and e respectively, if Po = 10cm, ﹤ APB = 40 ° Find: (1) the circumference of △ ped; (2) the degree of ∠ doe