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It is known that P is a point AC on the diagonal of square ABCD (not coincident with a and C), PE is perpendicular to point E, PF is perpendicular to point F 1, and CD is perpendicular to point F 1 It is known that P is a point AC (not coincident with a and C) on the diagonal of square ABCD, PE is perpendicular to point E, PF is perpendicular to point F 1. Verify BP = DP 2. If the quadrilateral PECF rotates counterclockwise around point C, is there BP = DP in the process of rotation? If so, please prove it, if not, please use counter examples Try to select two vertices of square ABCD and connect them with the two vertices of quadrilateral PECF, so that the length of the two segments is equal when the quadrilateral PECF rotates counterclockwise around point C, and prove your conclusion Thank you on your knees
The first question should be able to do, two feet 45 degrees, vertical is a square, very good proof
Second, no, take a counter example. For example, if P falls on BC, it should not be difficult to prove that PECF is a square
The third one is not very clear. It seems that it's easy to make mistakes. If you look at it again, it's mainly PECF or square. If you think about it carefully, it shouldn't be difficult. If you're satisfied, you can give it to me. Sure
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- 1. The side length of square ABCD is a, PA ⊥ plane ABCD, PA = a, m, n are the midpoint of PD, Pb respectively, then the cosine value of the angle formed by the line am and cn is
- 2. As shown in the figure, a point P, PE ⊥ ad in the square ABCD is equal to E. if Pb = PC = PE = 5, then the side length of the square is______ ．
- 3. It is known that: as shown in the figure, in square ABCD, point E is a point on the side of AD, and the intersection diagonal of CE is BD at point P, PE = AE. (1) verification: CE = 2ed. (2) when Pb = 6cm Please teach me the way I understand, Just a second question
- 4. Let p be any point on the inferior arc ad of circumscribed circle of square ABCD, then the ratio of PA + PC to Pb is______ ．
- 5. The center of square ABCD is O, the area is 1989cm2. P is a point in the square, and ∠ OPB = 45 °, PA: Pb = 5:14=______ ．
- 6. There is a point P in the square ABCD with side length 2. Find the minimum value of PA + Pb + PC. please write the procedure. 7. Ab There is a point P in the square ABCD with side length 2. Find the minimum value of PA + Pb + PC. please write the procedure 7. AB and AC are the diameter chord of circle O respectively, D is the point on inferior arc AC, De is perpendicular to ab at point h, intersecting circle O at point E, intersecting circle AC at point F, P is a point on extended line of ED
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- 8. In rectangular ABCD, the angle DEA = 30 degrees, e is a point on CD, and AE = AB = 8cm, find (1) the degree of angle EBC; (2) the area of triangle ade
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- 10. The height of a trapezoid is 7 cm. The product of it and the upper bottom is 78.4, and the product of it and the lower bottom is 178.4. So the area of this trapezoid is______ Square centimeter
- 11. As shown in Figure 1, it is known that P is a point on the diagonal AC of square ABCD (not coincident with a and C), PE ⊥ BC at point E, PF ⊥ CD at point F. (1) verification: BP = DP; (2) as shown in Figure 2, if quadrilateral PECF rotates counterclockwise around point C, is there always BP = DP in the process of rotation? If so, please prove it; if not, please explain it with counter examples; (3) try to select two vertices of square ABCD and connect them with the two vertices of quadrilateral PECF respectively, so that the length of the two segments is always equal when quadrilateral PECF rotates counterclockwise around point C, and prove your conclusion
- 12. As shown in Figure 1, it is known that P is a point on the diagonal AC of square ABCD (not coincident with a and C), PE ⊥ BC at point E, PF ⊥ CD at point F. (1) verification: BP = DP; (2) as shown in Figure 2, if quadrilateral PECF rotates counterclockwise around point C, is there always BP = DP in the process of rotation? If so, please prove it; if not, please explain it with counter examples; (3) try to select two vertices of square ABCD and connect them with the two vertices of quadrilateral PECF respectively, so that the length of the two segments is always equal when quadrilateral PECF rotates counterclockwise around point C, and prove your conclusion
- 13. As shown in the figure, P is a point on the diagonal AC of ABCD with side length 1 (P does not coincide with a and C), and point E is on the ray BC, and PE = Pb. (1) verification: PE = PD; (2) PE ⊥ PD
- 14. As shown in the figure, P is a moving point on the diagonal AC of the square ABCD with side length 1 (P does not coincide with A. C), and point E is on the ray BC And PE = PB Verification (1) PE vertical PD Let AP = x △ PBE area be y Find out the function relation between Y and X, and write out the value range of X
- 15. In square ABCD, point P is a point on edge CD, and PE ⊥ dB, PF ⊥ Ca, perpendicular feet are points E and f respectively, What is the quantitative relationship between PE + PF and the diagonal length of a square
- 16. It is known that, as shown in the figure, P is a point inside the square ABCD, and there is a point e outside the square ABCD, satisfying that the angle Abe = angle CBP, be = BP, If PA: Pb = 1:2, angle APB = 135 degrees, what is the chord value of angle PAE?
- 17. As shown in the figure, P is a point in the square ABCD, and there is a point e outside the square ABCD, which satisfies ∠ Abe = ∠ CBP, be = BP,
- 18. Let p be a point in a square ABCD, △ ABP is congruent with △ CBE. If BP = a, let PE be the area of the square with side length
- 19. For a point in the square ABCD, rotate △ ABP clockwise 90 ° around B to the position of △ CBE, if BP = A. calculate the area of the square with PE as the side length Write every step in detail Using Pythagorean theorem
- 20. As shown in the figure, point P is a point in square ABCD. Rotate △ ABP around point B and coincide with △ BCP '. If BP = 3, find the length of PP ′ De