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In square ABCD, point E.F is on edge BC and CD respectively. If AE = 4, EF = 3 and AF = 5, what is the area of square ABCD
This is very simple! Because AE = 4, EF = 3, AF = 5, the triangle AEF is a right triangle. It is easy to know that the triangle Abe is similar to the triangle ECF, then AB / EC = AE / EF = 4 / 3, so EC is 3 / 4 of AB, be is 1 / 4 of ab. let AB = x, be = x / 4, the square of X + (x / 4) = 4, the square of 4 = 16, the square of x = 256 / 17, that is, the square area is 256 / 17
As shown in the figure, in square ABCD, points E and F are on edge BC and CD respectively. If AE = 4, EF = 3 and AF = 5, then the area of square ABCD is equal to ()
A. 22516B. 25615C. 25617D. 28916
∵ AE = 4, EF = 3, AF = 5 ∵ AE2 + ef2 = af2, ∵ AEF = 90 ∵ AEB + ∵ FEC = 90 ∵ square ABCD ∵ Abe = ∵ FCE = 90 ∵ CFE + ∵ CEF = ∵ EAB + ∵ AEB = 90 ∵ FEC = ∵ EAB ∽ ECF ∵ EC: ab = EF: AE = 3:4, namely EC = 34ab = 34bc ∵ be = bc4 = ab4 ∵
As shown in the figure, in square ABCD, h is on BC, EF ⊥ ah intersects AB at point E and DC at point F. if AB = 3 and BH = 1, the length of EF is obtained
If FM ⊥ AB is perpendicular to m, then FM = AB = 3. (2 points) ≁ 1 + ≌ RT △ ABH (2 points) ≌ EF = ah = AB2 + BH2 = 32 + 12 = 10 (2 points)
In a pyramid p-abcd, planar pad ⊥ planar ABCD, ab ∥ CD, △ pad are equilateral triangles. It is known that BD = 2ad = 8, ab = 2dc = 4 √ 5
(1) Let m be a point on PC, and prove that planar MBD ⊥ planar pad
(2) Finding the volume of a pyramid p-abcd
1) According to Pythagorean theorem, △ ADB is a right triangle, and angle ADB is a right angle, that is BD ⊥ ad,
From planar pad ⊥ planar ABCD, we can get: BD ⊥ planar pad,
Because BD is on planar MBD, planar MBD ⊥ planar pad
2) Area of quadrilateral ABCD = area of △ ADB + area of △ CDB
The height of △ CDB is the same as that of △ ADB, while the bottom AB = 2dc,
So the area of △ CDB = half of the area of △ ADB
So the area of quadrilateral ABCD = △ ADB × 1.5 = 4 × 8 △ 2 × 1.5 = 24
The height of pyramid p-abcd = 2 √ 3
So the volume of p-abcd is 24 × 2 √ 3 △ 3 = 16 √ 3
It is known that in p-abcd, the bottom surface ABCD is rectangle, ad = 2, ab = 1, PA is vertical plane ABCD,
E. F is the midpoint of line AB and BC respectively. (1) prove that PF is perpendicular to fd. (2) judge whether there is a point G on PA such that eg / / plane PFD, prove your conclusion. (3) if the angle between Pb and plane ABCD is 45 degrees, find the remainder of dihedral angle a-pd-f
(1) If AF is vertical to FD and a is p projected on the plane ABCD, PF is vertical to fd
(2) Take the midpoint of Da, DP and AP as O, m and g respectively. It is easy to prove that the quadrilateral mgbf is rectangular, then eg / / plane PFD,
(3) If the angle onf is calculated, the cosine value is 3 / (3) under the root sign
As shown in the figure, the bottom of the pyramid p-abcd is a rectangle, and the side pad is an equilateral triangle. When the ad / AB value is, Pb ⊥ AC
For example. Pad ⊥ ABCD., for Pb ⊥ AC, as shown in the figure, PE ⊥ ad, thus PE ⊥ ABCD.PB ⊥ AC. & nbsp;; EB ⊥ AC (three vertical lines) ⊥ EAB ⊥ ABC. (AD / 2) / AB = AB / BC = AB / AD, 2Ab & amp; sup2; = ad & amp; sup2
It is known that in trapezoidal ABCD, AD / / BC, ∠ ABC = 60, BD bisects ∠ ABC, and BD ⊥ DC
(1) Verification: trapezoid ABCD is isosceles trapezoid
(2) When CD = 1, find the circumference of ladder ABCD
In BDC,
As shown in the figure, it is known that in the trapezoidal ABCD, ad ∥ BC, ∠ ABC = 60 ° BD bisects ⊥ ABC, and BD ⊥ DC. (1) prove that the trapezoidal ABCD is isosceles trapezoid; (2) when CD = 1, find the perimeter of isosceles trapezoid ABCD
(1) It is proved that: ∵ BD bisection ∵ ABC, ∵ abd = ∵ CBD, ∵ ABC = 60 °, ∵ CBD = 30 °, ∵ BD ⊥ DC, ∵ BDC = 90 °, ∵ C = 60 °, ∵ trapezoid ABCD is isosceles trapezoid; (2) through point D, make de ∥ ab, ∵ ad ∥ BC, ∵ quadrilateral abed is parallelogram, ? CD = 1, ∥ BC = 2
In the parallelogram ABCD, take a point P, BP = 1, AP = 4, PC = 5 to find PD?
Because in the parallelogram ABCD, AB / / CD, AD / / BC, agpe, gbfp, pfch, ephd are all parallelograms, so AE = BF = PG, ed = FC = pH, angle AEP = angle PHC = angle ADC, angle BFP = angle ped = angle BCD
The parallelogram ABCD has a point P beside ad, AP vertical PC, Pb vertical PD. It is proved that ABCD is a rectangle
Certification:
Connect AC and BD at point o
If Po is connected, Ao = Co, Bo = do
∵∠APC=90°,AO=CO
∴AC=2AO
Similarly, we can get BD = 2ao
∴AC=BD
The parallelogram ABCD is a rectangle
RELATED INFORMATIONS
- 1. As shown in the figure, in the square ABCD, ab = 1, point P is a point on the diagonal AC, and AP and PC are used as the diagonal to make the square respectively, then the sum of the perimeter of the two small squares is______ ．
- 2. It is known that, as shown in the figure, P is a point in the square ABCD, and there is a point e outside the square ABCD, satisfying the angle Abe = angle CBP, be = BP 2.PB⊥BE.
- 3. As shown in the figure, in rectangular ABCD, e is a point on CD, angle DEA = 30 degrees, and AE = AB = 8cm, find 1, degree of angle EBC, 2, area of triangle Abe
- 4. The ratio of the top to the bottom of a trapezoid is 3:5. The top is 1 meter shorter than the bottom, and the bottom is 25% higher. The area of the trapezoid is ()
- 5. As shown in the figure, in the diamond ABCD, ab = 2, ∠ C = 60 °, the diamond ABCD rolls to the right on the straight line l without sliding. Each rotation of 60 ° around a vertex is called an operation. After 36 such operations, the total length of the path that the diamond center o passes through is (the result retains π)______ ．
- 6. In the right angle trapezoid ABCD, AD / / BC, ∠ B = 90 °, ad = 24cm, BC = 26cm, the moving point P moves rapidly from a along the side of ad to D at the speed of 1cm per second In the right angle trapezoid ABCD, AD / / BC, ∠ B = 90 °, ad = 24cm, BC = 26cm, the moving point P starts from a and moves at the speed of 1cm / s along the ad side to D, the moving point Q starts from point C and moves at the speed of 3cm / s along the CB side to B, and P and Q start from points a and C at the same time. When one point reaches the end point, the other point also stops moving?
- 7. As shown in the figure, in the known trapezoidal ABCD, ad ‖ BC, ∠ B = 30 °, C = 60 °, ad = 4, ab = 33, then the length of the bottom BC is______ ．
- 8. As shown in the figure, a, B, C and D are the four vertices of rectangle ABCD, ab = 25cm and ad = 8cm. The moving points P and Q start from point a and C at the same time. Point P moves to point B at the speed of 3cm / s until point B, and point Q moves to point d at the speed of 2cm / S. (1) P and Q, from the start to the second, PQ ‖ ad? (2) Question: P, Q two points, from the start to a few seconds, quadrilateral pbcq area of 84 square centimeters?
- 9. The height of a trapezoid is 10 cm. If the upper bottom is increased by 6 cm and the lower bottom is increased by 4 cm, it becomes a parallelogram. Calculate the area of the original trapezoid
- 10. The top of a trapezoid is three times as big as the bottom. If the bottom is extended by 6 cm, it will become a parallelogram. How many cm are the top and bottom of the trapezoid?
- 11. Let the vertex a of the square ABCD be the plane ABCD, PA = AB = a, and find the dihedral angle between the plane PAB and the plane PCD
- 12. It is known that PA is perpendicular to the plane of rectangle ABCD, and M is the midpoint of ab. if PD and ABCD form an angle of 45 degrees, it is proved that plane PCM is perpendicular to plane PCD
- 13. What do QA and QC mean?
- 14. As shown in the figure, place a rectangle in the plane rectangular coordinate system, OA = 2, OC = 3, e is the midpoint of AB, and the image of inverse scale function passes through point E and connects with BC Intersection at point F (1) Finding the analytic formula of straight line OB and the analytic formula of inverse scale function (2) Connect of and OE, point P is on the straight line ob, and s triangle AOP = 2S quadrilateral oebf, calculate the coordinates of point P
- 15. As shown in the figure, two pieces of paper of equal width are crossed and overlapped, and the quadrilateral ABCD of the overlapped part is______ Shape
- 16. In the plane rectangular coordinate system, the coordinates of the three vertices of square ABCD are a (0,0), B (- 2,0), D (0,2), then the coordinates of point C are Then the coordinates of the midpoint of the square BC are 2. If point P (a, 3-2a) is on the bisector of the second and fourth quadrants, then the value of a is
- 17. Given that the rectangle ABCD, and ab = 5, ad = 3, establish an appropriate plane rectangular coordinate system, find the coordinates of each vertex of the rectangle
- 18. If the inner angle of the diamond is 120 ° and the length of its diagonal is 12cm, the perimeter of the diamond is The answer is 48CM or 2 √ 3, remember that there are two cases
- 19. As shown in the figure, fold the rectangular ABCD paper along EF so that point d coincides with the midpoint d 'of BC side. If BC = 8, CD = 6, CF=______ ．
- 20. If the coordinates of the point a symmetric about the Y axis are (4, - 3), then the coordinates of the point a symmetric about the origin are (4, - 3)