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As shown in the figure, a, B and C are three points that are not on the same straight line, AA ′‖ BB ′‖ CC ′, and AA ′ = BB ′ = CC ′. Prove: plane ABC ‖ plane a ′ B ′ C ′
It is proved that: ∵ AA ′ = BB ′, AA ′‖ BB ′, ∵ a ′ B ′‖ AB are parallelogram, ∵ a ′ B ′‖ AB, ab ⊂ face ABC ∩ a ′ B ′‖ face ABC, same as B ′ C ′‖ face ABC, ∩ a ′ B ′ C ′ = B ′, ∩ face ABC ‖ face a ′ B ′ C ′
As shown in the figure, in RT △ ABC, ∠ ABC = 90 ° Ba = BC = 2, make the vertical lines AA ′ and CC ′ of plane ABC through a and C respectively, AA ′ = 2, CC ′ = 1, connect a ′ C and AC ′ at the point P, M is the point on the edge of BC, CM = 23. (I) prove: Line PM ‖ plane a ′ AB; (II) find the angle between line MP and plane a ′ AC
(1) It is proved that: ∵ AA ′⊥ plane ABC, ∵ CC ′⊥ plane ABC, ∵ AA ′∥ CC ′, ∵ a ′ PPC = & nbsp; a ′ AC ′ C & nbsp; = & nbsp; 21. Also ∵ cm = 23, BC = 2, ∵ BMMC = 2, ∵ PM ∥ a ′ B. also a ′ B ⊂ plane AA ′ B, PM is not in plane AA ′ B. (II) from (I), the angle between PM ∥ a ′ B, ∥ PM and plane AA ′ C is the angle between a ′ B and plane AA ′ C. let the midpoint of AC be o, in RT △ ABC, Ba = BC = 2, ∥ Bo ⊥ AC, and Bo = 2. ∵ AA ′⊥ surface ABC, ∩ AA ′⊥ surface ABC, ∩ AA ′∩ surface ABC = AC, ∩ Bo ⊥ surface ABC, ∩ BA ′ o is the angle between the straight line MP and the plane a ′ AC. ∵ Ao = 2, a ′ a = 2, ∩ a ′ o = 6, ∩ Tan ∠ BA ′ o = boa ′ = 33, ∩ BA ′ o = 30 °. Therefore, the angle between the straight line MP and the plane a ′ AC is 30 °
As shown in the figure, O is the center of gravity of △ ABC. The extension lines of AO and Bo intersect BC and AC at points E and D respectively. If AB = 12, the de length is ()
A. 3B. 4C. 6D. 8
∵ o is the center of gravity of △ ABC, ∵ ad = CD, be = CE, ∵ De is the median line of △ ABC, ∵ de = 12ab = 6
It is known that, as shown in the figure △ ABC, the intersection point of its bisector BD and CE is O, connecting Ao, and then using a protractor to check that the size relationship between ∠ Bao and ∠ Cao is ()
Equal
RELATED INFORMATIONS
- 1. O is the outer center of the triangle ABC. If vector Ao multiplies vector AB = vector Bo multiplies vector BC, the shape of the triangle ABC can be helped by God
- 2. In △ ABC, the degrees of a and B are as follows, and () A. ∠A=50°，∠B=70°B. ∠A=70°，∠B=40°C. ∠A=30°，∠B=90°D. ∠A=80°，∠B=60°
- 3. As shown in the figure, it is known that P is a point on the side BC of △ ABC, and PC = 2PB. If ∠ ABC = 45 ° and ∠ APC = 60 °, find the size of ∠ ACB
- 4. In the triangle ABC, f is a point on the edge of BC, and PC = 2PB,
- 5. In the triangle ABC, ab = AC = 2A, ∠ ABC = ACB = 15 ° CD is the height on the waist ab
- 6. If the sum of three numbers is equal to 18 and the sum of their squares is equal to 116, then the three numbers are equal______ ．
- 7. If the sum of three numbers is equal to 18 and the sum of their squares is equal to 116, then the three numbers are equal______ ．
- 8. The sum of squares of four numbers is 94. The product of the first number and the fourth number is 18 less than the product of the second number and the third number
- 9. There are two arithmetic sequences 2,6,10 190 and 2,8,14 A new sequence is composed of the common terms of the two arithmetic sequences from small to large, then the sum of the items of the new sequence is__________ .
- 10. There are two arithmetic sequences 2, 6, 10 , 190 and 2, 8, 14 The common terms of the two arithmetic sequences form a new sequence from small to large, and the sum of each item of the new sequence is calculated
- 11. ABC + CDA = ABCD ask what are ABCD In ABC, a is a hundred, B is a ten, C is a single, in CDA, C is a hundred, D is a ten, a is a single, in ABCD, a is a thousand, B is a hundred, C is a ten, D is a single
- 12. Δ ABC ≌ Δ CDA, AB and CD, BC and DA are corresponding edges. Write other corresponding edges and corresponding angles
- 13. In RT △ ABC, ∠ C = 90 degree, ∠ a = 30 degree, then BC: AC: ab = what
- 14. As shown in the figure, in the triangle ABC, the angle ABC is 90 degrees and the angle c is 30 degrees. Rotate the triangle ABC counterclockwise around point a to the position of triangle ab'c ', and point B falls on AC The value of tancdb 'is?
- 15. In the triangle ABC, the angle BAC = 90 ° AB = AC, take a point D in the triangle to make the angle abd = 30 ° and BD = Ba, AE perpendicular to BD and e to find the bisector angle EAC of AD
- 16. As shown in the figure, D in △ ABC is any point on edge BC, de and DF are bisectors of △ ADB and △ ADC respectively, connecting EF. Try to judge the shape of △ def and explain the reason
- 17. It is known that △ ABC and △ ade are equilateral triangles, and points a and E are on the same side of BC (1) As shown in Figure 1, if point D is on BC, write the quantitative relationship among line segments AC, CD and CE, and prove it; (2) as shown in Figure 2, if point D is on the extension line of BC, and other conditions remain unchanged, directly write the quantitative relationship among AC, CD and CE
- 18. In equilateral triangle ABC, is de / / BC. Angle ade an equilateral triangle?
- 19. Triangle ABC is an equilateral triangle, PA is perpendicular to plane ABC, and D is the midpoint of BC
- 20. Take any two numbers P and Q (P ≠ q) from the four numbers 2, 3, 4 and 5 to form the functions y = px-2 and y = x + Q, and make the image of these two functions If the intersection point is on the right side of the straight line x = 3, how many pairs of ordinal numbers are there in total?