image imagewidth (px) 5 3.56k | id stringlengths 36 36 | question stringlengths 56 1.48k | answer stringlengths 1 672 |
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11be94d6-b686-4613-8b1b-38879a2901b0 | Executing the algorithm flowchart as shown in the figure, the final output $$y$$ equals ___. | $$63$$ | |
a654adf3-5c56-4afb-a48d-9345e0f36bbb | As shown in the figure, it is known that all the side lengths of tetrahedron $\text{ABC}-{{\text{A}}_{1}}{{\text{B}}_{1}}{{\text{C}}_{1}}$ are 1, and $\text{A}{{\text{A}}_{1}}\bot $ the base ABC. Find the volume of the tetrahedron ${{\text{B}}_{1}}-\text{AB}{{\text{C}}_{1}}$. | $\frac{\sqrt{3}}{12}$ | |
14097ea6-d34b-4ac5-a1a5-edb8f06a81d2 | As shown in the figure, in the Cartesian coordinate system, the line $$l$$:$$y=x+2$$ intersects the $$x$$-axis at point $$A$$ and intersects the $$y$$-axis at point $$A_{1}$$. Points $$A_{2}$$, $$A_{3}$$, $$\cdot\cdot\cdot$$ lie on line $$l$$. Points $$B_{1}$$, $$B_{2}$$, $$B_{3}$$, $$\cdot \cdot\cdot$$ are on the posi... | $$2^{n+1}-2$$ | |
889375c7-6860-463f-baa5-308e90b108fb | Given that the distance between locations A and B is 4 kilometers. At 8:00 AM, Person A departs from location A to walk to location B, while at 8:20 AM, Person B departs from location B to ride a bicycle to location A. The graph shows the relationship between the distance (in kilometers) from location A and the time (i... | 8:40 | |
d1b6d3be-0b29-4ea0-a1f0-8c9543af9e80 | As shown in the figure, in triangle ABC, S$_{\triangle ABC}$=36, DE∥AC, FG∥BC, points D, F are on AB, E is on BC, and G is on DE, and BF=FD=DA, then S$_{quadrilateral BEGF}$=? | 12 | |
8eb7f26b-2087-47af-aee9-84d384471c86 | As shown in the figure, in the right triangle $\triangle ABC$, $\angle B={{90}^{\circ }}$, and $ED$ is the perpendicular bisector of $AC$, intersecting $AC$ at point $D$ and $BC$ at point $E$. Given $\angle C={{40}^{\circ }}$, find the measure of $\angle BAE$. | 10 | |
4133f371-1f0b-4f98-8b2c-bb83d4f35224 | Fold a rectangular sheet of paper as shown in the figure, with BD and BE as the folding creases. If ∠ABE = 20°, then find ∠DBC in degrees. | 70$textdegree$ | |
8e4ec53c-9ad2-4617-8964-a59a782b5dde | Input $$x=5$$, after running the following program, $$y$$ equals ___. | $$16$$ | |
f3417ea0-6278-41e9-b2ac-c348a83e3f9b | Write down the results of the following pseudo-code. If $$x=10$$, then $$p=$$___; If $$x=-1$$, then $$p=$$___. | $$1$$ $$-1$$ | |
747e9edd-7856-4a47-a858-06b39dffea2c | Observe the following patterns: ${{7}^{1}}=7$, ${{7}^{2}}=49$, ${{7}^{3}}=343$, ${{7}^{4}}=2041$, ${{7}^{5}}=16807$, ${{7}^{6}}=117649$, Based on the patterns in the above formulas, what do you think is the last digit of ${{7}^{2018}}$? | 9 | |
f4263569-d952-4a54-aaa5-9d68c9232f25 | On a certain spring weekend, the sun was shining brightly, making it perfect for outdoor activities. In the afternoon, Xiao Ming and Xiao Jing, who lived in the same community, both coincidentally decided to walk along the same route to a park to admire the flowers. Xiao Ming set off 5 minutes earlier than Xiao Jing, a... | 720 | |
e6824688-3645-4637-bc13-4d97b0c4c93e | The graph of the function is shown in the figure above. Regarding this function, the correct conclusion is ______ (fill in the serial number). 1. The graph of the function is an axis-symmetric figure; 2. The graph of the function is a point-symmetric figure; 3. When x > 0, the function has a minimum value; 4. The poin... | 1. 2. When x changes to -x, y changes to -y. Thus, the point (x, y) corresponds to (-x, -y), indicating that the graph of the function is a point-symmetric figure and not an axis-symmetric figure. Therefore, 2 is correct and 1 is incorrect; 3. When x > 0, the graph of the function has a lowest point, so the function ha... | |
01b79c1f-ff33-407e-b024-a9b95c7e21bc | In a plane rectangular coordinate system, if the coordinates of point $$P(x,y)$$, $$x$$ and $$y$$, are both integers, then point $$P$$ is called a lattice point. If all the vertices of a polygon are lattice points, then the polygon is called a lattice point polygon. The area of a lattice point polygon is denoted as $$S... | (1)$$3$$,$$1$$,$$6$$ (2)$$79$$ | |
ef3c8e67-7441-4a1b-959b-d60bfa4fd1ca | As shown in the figure, in $\Delta ABC$, $M$ is an arbitrary point on $BC$ but different from $B$ and $C$. Point $N$ satisfies ${AN}^\to = 2 {NM}^\to$, if ${AN}^\to = x {AB}^\to + y {AC}^\to$, then the minimum value of $x^2 + 9 y^2$ is | $\frac{2}{5}$. | |
054f202b-8617-4aea-b1a3-a486ef080234 | As shown in the figure, given the quadrilateral $ABCD$, AC is the perpendicular bisector of BD, the foot is E, O is a point outside the line BD, and it is known that the magnitudes $\left| \overrightarrow{OB} \right|=5$, $\left| \overrightarrow{OD} \right|=3$. Then $\left( \overrightarrow{OA}+\overrightarrow{OC} \right... | 16 | |
8eacf5cd-3a4b-484e-8853-798c1f6609a9 | The parabola ${{x}^{2}}=2py(p > 0)$ has its focus at , and its directrix l intersects with the curve at points A and B. If is an equilateral triangle, then equals. | $3\sqrt{2}$ | |
94068071-040b-4ebf-9ce0-0cc050dcc2fe | As shown in the figure, in a plane Cartesian coordinate system, point M is on the positive half of the x-axis. Line l passes through point M and is parallel to the y-axis. Line l intersects the curves of the inverse proportional functions y=\frac{8}{x} (x>0) and y=\frac{k}{x} (x>0) at points P and Q, respectively. If t... | -20 | |
5a6d15a4-1b7c-47ed-bfbc-8a2d279defe5 | The three-dimensional view of a triangular prism is shown in the figure below. The surface area of the circumscribed sphere of this triangular prism is ?. | $6\pi $ | |
eaecf148-775c-4b67-9dc5-3b83b3b5226c | As shown in the figure, after the fan-shaped paper fan is fully opened, the angle between the two bamboo strips $$AB$$ and $$AC$$ is $$120^{ \circ }$$, the length of $$AB$$ is $$ \quantity{30}{cm}$$, and the non-pasted part $$AD$$ is $$ \quantity{10}{cm}$$ long. Hence, the area of the pasted part is equal to ___$$\unit... | $$\dfrac{800}{3}\pi $$ | |
063743e1-0a2e-491f-a22d-86048bee5a35 | A rectangular wooden board has a circular hole in its upper right corner. Imagine transforming it into a tabletop for a hot pot, with the board's size remaining unchanged, and the circle's center located at the intersection of the diagonal lines of the rectangular tabletop. A carpenter devised a clever solution: after ... | $$\quantity{18}{cm}$$,$$\quantity{31}{cm}$$ | |
9139836b-1fc6-4930-ac07-08cf35d1181e | In quadrilateral ABCD, points E, F, G, H are the midpoints of AB, BC, CD, AD respectively. As shown in the figure, if AC = BD, then the shape of quadrilateral EFGH is ______. | rhombus. | |
5e213d9b-63f9-4d71-9c9d-89004bd2a002 | Given that the distribution of the random variable $$\xi$$ is as follows: find the minimum value of $$E( \xi )$$ as ___. At this time, $$b=$$___. | $$\dfrac{3}{4}$$ $$\dfrac{1}{2}$$ | |
ae36c4c7-4670-4a04-9bab-9f8ce36951a7 | As shown in the figure, in the rectangular prism $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$, $AB=\sqrt{3}AD=\sqrt{3}A{{A}_{1}}=\sqrt{3}$, point $P$ is a moving point on the line segment ${{A}_{1}}C$ (including endpoints of the segment). Determine whether the following statements are correct.
1. When $\overrightarrow{... | 1.
2. | |
db47c938-17be-48ea-977b-922bc49d59b6 | As shown in the figure, if ∠1 = 45°, ∠2 = 120°, then the measure of an angle corresponding to ∠3 is ______, the measure of the interior opposite angle to ∠3 is ______, and the measure of the co-interior angle to ∠3 is ______. | 60°; 60°; 120°. | |
a8a97146-89c7-41ab-80c1-6dabbaba7444 | As shown in the figure, in the square ABCD, △ABE is an equilateral triangle, connect DE, CE, extend AE to intersect CD at point F, then the degree of ∠DEF is: | 105° | |
f6928ae9-6b4e-458a-8e66-329c1c47d270 | As shown in the figure, it is known that $AB\bot BC,EF\bot BC,CD\bot AD$, if $AB=CD=2cm$, $AE=3cm$, then the area of \( \triangle AEC \) is cm$^{2}$. | 3 | |
1ba7b392-ebe2-45f0-ada1-b74247523f4d | In the right figure (unit: cm), if a rectangle is divided into three shapes: $$A$$, $$B$$, $$C$$, then the ratio of the areas of $$A$$, $$B$$, and $$C$$ is ______. | 1:3:2 | |
c95dcbb4-5933-4d98-96db-6a8111fae15f | As shown in the figure, the graphs of the functions y = -x and y = -$\frac{4}{x}$ intersect at points A and B. Vertical lines pass through points A and B, meeting the y-axis at points C and D respectively. What is the area of the quadrilateral ACBD? | 8 | |
86e72e32-1050-4021-9283-34195da1bfc2 | As shown in the figure, point $A$ is on the hyperbola $y=\frac{2\sqrt{3}}{x}\left( x > 0 \right)$, point $B$ is on the hyperbola $y=\frac{k}{x}\left( k > 0 \right)$ (point $B$ is to the right of point $A$), and $AB\parallel x$-axis. If the quadrilateral $OABC$ is a rhombus and $\angle AOC=60{}^\circ$, then $k=$. | $6\sqrt{3}$. | |
a39d7608-b998-44c9-9e26-43d2eb2b5bff | As shown in the figure, in rectangle ABCD, point E is the midpoint of side AD. BE intersects diagonal AC at point F. Find the area ratio of $\vartriangle AFE$ to $\vartriangle BCF$. | $\frac{1}{4}$ | |
d82d2ade-02cc-4c1a-a49c-3dce34799d77 | Arrange 2017 squares with side length 2 as shown in the figure, where O$_{1}$, O$_{2}$, O$_{3}$, O$_{4}$, O$_{5}$, … are the intersections of the diagonals of the squares. What is the sum of the areas of the shaded regions? | 2016 | |
90470886-540d-4d9d-806c-6d622a4a849f | As shown in the figure, $$OA$$ represents ___ deviation ___ $$28^{\circ}$$ direction, ray $$OB$$ represents ___ direction, $$\angle AOB=$$___. | north east southeast $$107^{\circ}$$ | |
9384948c-c410-4151-9d39-9739761fef34 | Observe the diagram. The sum of the numbers in the nth row is equal to 2017$^{2}$. | 1009 | |
ae38e67e-ce3e-4cf9-ac60-e0f64ef38b5c | As shown in the figure, $$AB$$ and $$CD$$ are chords of circle $$\odot O$$, and $$AB\perp CD$$. $$BE$$ is a diameter of circle $$\odot O$$. If $$AC=3$$, then $$DE=$$______. | $$3$$ | |
573a5ba8-9683-412d-83e7-081f9c4fdb83 | If the equation has an extraneous root, then the value of m is ______. | -3 | |
6611106d-22f1-4dce-abc8-57794d9bed5d | In circle ⊙O, AB is the diameter of ⊙O, and E, F are the trisection points of arc BC. ∠BOF=35°, then ∠AOC=. | 75° | |
639803d3-f9a8-4f52-a639-05fd00b96286 | There is a clamp as shown in the figure, AB=2BC, BD=2BE. In front of the clamp there's a rectangular solid object with thickness PQ=6cm. If you want to use the sharp points A and D of the clamp to hold the points P and Q, then the minimum distance EC that the clamped part EC needs to be opened is how many cm? | 3 | |
48b30072-4901-45e4-9825-970730057f9a | As shown in the figure, a skier slides from point $$A$$ to point $$B$$ along a slope with an inclination angle of $$34^{\circ}$$. Given that $$AB=500$$ meters, the vertical drop of the skier is ___ meters (reference data: $$\sin 34^{\circ}\approx 0.56$$, $$\cos 34^{\circ}\approx 0.83$$, $$\tan 34^{\circ}\approx 0.67$$)... | $$280$$ | |
5cf84e5f-5478-47b4-9463-2a8611dc7005 | Given that the base $ABCD$ of the quadrangular prism $ABCD-{{A}_{1}}B{{C}_{1}}{{D}_{1}}$ is a rectangle, with $AB=5$, $AD=3$, $A{{A}_{1}}=4$, and $\angle BA{{A}_{1}}=\angle DA{{A}_{1}}=60{}^\circ $, find $A{{C}_{1}}=$. | $\sqrt{82}$ | |
d05ba3dd-76db-406f-b606-023c7ed395ec | As shown in the figure, in $\vartriangle \text{ABC}$, the perpendicular bisector DE of AC intersects AB at point E and intersects AC at point D. Connect CE. If $\angle \text{A}={{34}^{\circ }}$ and $\angle \text{ACB}={{76}^{\circ }}$, then $\angle \text{BCE}=$ . | 42°. | |
7a04061e-f9f4-4512-a702-7392704b901d | As shown in the figure, connect the midpoints of each side of quadrilateral ABCD to obtain quadrilateral EFGH. By adding a condition, we can ensure that quadrilateral EFGH is a rhombus. | AC=BD | |
5bc2246a-d8e9-43a0-b92d-a906bea31fb5 | If real numbers $$x$$ and $$y$$ satisfy: $$\left \lvert x\right \rvert > \left \lvert y\right \rvert $$, then we say $$x$$ is farther away from $$0$$ than $$y$$. As shown in the figure, it is known that points $$A$$, $$B$$, $$C$$, $$D$$, and $$E$$ on the number line correspond to the real numbers $$a$$, $$b$$, $$c$$, $... | $$0$$ | |
bfbcf47f-876c-4764-ab62-ec2306f5bd5c | Given the functions $f(x)$ and $g(x)$ are provided in the following table. Find the value of $f[g(1)]. | 2 | |
b56af965-e5ee-4b56-b42c-fd3091195923 | As shown in the figure, in right triangle ABC, ∠ACB = 90°, AC = BC. With A as the center of a circle, the arc DF of length AD as the radius intersects the extension line of AC at F. If the areas of the two shaded parts in the figure are equal, then $\frac{\text{AD}}{\text{DB}}=$. | $\frac{\sqrt{2\pi }+2}{\pi -2}$ | |
4fa688ec-305f-441d-b61a-a78b4c2c2b87 | As shown in the figure, in the parallelogram $$ABCD$$, vertices $$A$$, $$B$$, and $$D$$ are on the coordinate axes, $$AD=5$$, $$AB=9$$, and the coordinates of point $$A$$ are $$(-3,0)$$. Find the coordinates of point $$C$$. | $$(9,4)$$ | |
9c1156f8-f4eb-4faa-a837-ff0baec7af2f | As shown in the figure, BD and DE are the midlines of triangles ABC and ABD respectively. If the area of triangle ABC is 16 cm², then the area of triangle EBD is cm². | 4 | |
22ba1423-2c12-47c4-b9cc-3baa2562e047 | Given that the graph of the linear function y$_{1}$=kx+m and the quadratic function y$_{2}$=ax$^{2}$+bx+c is as shown in the figure, they have two intersection points A(1,1) and B(6,5). Then the range of values for the independent variable x that satisfies y$_{1}$ > y$_{2}$ is ______. | The range of values for the independent variable x that satisfies y$_{1}$ > y$_{2}$, as seen from the graph, is when the graph of the linear function is above that of the quadratic function. From the graph, it appears that the range is 1 < x < 6, where the linear function graph is above the quadratic function graph, so... | |
359a6890-152b-4e6b-a171-3ab9f65f7af1 | In quadrilateral $$ABCD$$, $$E$$ is a point on side $$BC$$. Connect $$AE$$ and it intersects $$BD$$ at point $$F$$. If $$EC=2BE$$ and $$EF=2$$, then the value of $$AE$$ is ______. | $$8$$ | |
78d48442-59eb-4576-98dc-51da659a5d85 | As shown in the figure, $D$ and $E$ are points on the sides $AB$ and $AC$ of $\Delta ABC$ respectively, and $\angle ADE=\angle ACB$. If $AD=2$, $AB=6$, $AC=4$, then $AE=$. | 3 | |
ed24d3d3-3f7b-4b3e-8311-3140e8e67a42 | If the solutions for the system of equations have x equal to y, then the value of k is. | 11 | |
20b5760c-f435-4cdf-9874-4829dd68f599 | As shown in the figure, in the rectangular coordinate system, given the points $A(-3$, $0)$, $B(0$, $4)$, the triangle $\Delta OAB$ is repeatedly rotated, resulting in $\Delta 1$, $\Delta 2$, $\Delta 3$, $\Delta 4$, …. Find the coordinates of the right angle vertex of $\Delta 301$. | (1200, 0) | |
deca180b-df0a-4085-a8d1-6df79d4f70b0 | In the figure, if $$ \angle BAD=75^{ \circ }$$, then $$ \angle BCD=$$___. | $$105^{ \circ }$$ | |
08da363d-3fe5-4569-a5f2-5d267d52f735 | As shown in the diagram, this is a flowchart of an algorithm. What is the final output value of $$W$$? | $$14$$ | |
bb8bb053-99e1-4f90-9972-d9764dfbf2d2 | As shown in the figure, the diagram is composed of a circle and an arc with the same radius. If a dart is thrown at the diagram, the probability that the dart lands in the shaded area is ___. | $$\dfrac{1}{3}$$ | |
a048541b-5c76-472c-94ab-426edd3075b9 | The table provides several sets of corresponding data for the production quantity $$x$$ (tons) and production energy consumption $$y$$ (tons of standard coal) recorded in the process of producing product $$A$$ at a factory after energy-saving and consumption-reduction technological transformation. If according to the d... | $$3$$ | |
d0ad81ff-c2e4-4374-a80e-6dbdf9f29de4 | The following is the ruler and compass construction process of "constructing a circle with a known line segment as the diameter". Given: As in figure 1, line segment $$AB$$. Required to construct: circle $$\odot O$$ with $$AB$$ as the diameter. Construction method: As in figure 2, (1) Construct arcs with centers at $$... | The criteria of perpendicular bisector and the definition of a circle | |
269bd4b5-6ed3-4330-86ac-9103c0482871 | In the plane rectangular coordinate system, the position of quadrilateral $$ABCD$$ is shown in the figure. If $$AB\bot AD$$, $$AB\parallel CD$$, and $$AB=5$$, and the coordinates of point $$A$$ are $$\left ( -2,7\right ) $$, then the coordinates of point $$B$$ are ___. | $$\left ( 3,7\right ) $$ | |
1a652296-ed4b-412e-9e0b-70cd071fe120 | Write the result of the following sequence execution. Input x=6, then p=_____; Input x=20, then p=_____. | 1;10.5 | |
dc06badc-ecd1-4e28-9b36-bc48a9fe1f8f | As shown in the figure, point C is on ⊙O. Rotate the central angle ∠AOB around point O in the counterclockwise direction to ∠A′OB′, with the rotation angle α (0° < α < 180°). If ∠AOB = 30°, ∠BCA′ = 20°, and the radius of ⊙O is 6, find the arc length of $\overset{}{\mathop{\text{AB }\!\!'\!\!\text{ }}}\,$. (Leave the re... | $\frac{10}{3}\pi $ | |
e8dc2950-263f-4557-9ac6-cc9e842df6ed | For a real number $$x$$, perform the operation according to the procedure shown in the diagram, with the rule: the operation runs from "input a real number $$x$$" to "is the result greater than $$88$$?" as one operation. If the operation stops after only one run, then the range of values for $$x$$ is ___. | $$x > 49$$ | |
b7e66b6e-bc8f-4204-91c5-10926340cea3 | As shown in the figure, it is known that $$AD=6$$, $$\angle CAD=\angle DBC$$, $$\angle BAD=\angle ABC$$, then $$BC=$$___. | $$6$$ | |
7d09fd43-1406-46a7-b274-5e8c35000d9e | As shown in the figure, point E is any point on the side AB of the parallelogram ABCD (not including endpoints A and B). Connect AC and CE. Construct a rhombus ECFG with CE as the side, and then connect BG. Given that AB = AC and \( \angle CEG = \angle CAB \), determine which of the following conclusions are necessaril... | 2.3.4. | |
5c7d2b07-f31c-44d8-8402-6d7a91dd9549 | As shown in the figure, in $$\triangle ABC$$, $$AB+AC=6cm$$, the perpendicular bisector of $$BC$$, line $$l$$, intersects $$AC$$ at point $$D$$. Therefore, the perimeter of $$\triangle ABD$$ is ___ $$cm$$. | $$6$$ | |
24f29277-c172-49c8-b7f5-e5465f2883b6 | As shown in the figure, the line y$_{1}$=-x+a and y$_{2}$=bx-4 intersect at point P. It is known that the coordinates of point P are (1, -3). Then the solution set of the inequality -x+a<bx-4 with respect to x is. | $x > 1$ | |
c9ea3765-59ee-413d-8e94-7a2332e00f1d | Given that the function $f(x)$ is a differentiable function defined on $R$, the line $y = kx + 2$ intersects with the graph of the function $f(x)$, as shown in the figure, then the equation of the tangent line at the point $(3, g(3))$ on the graph of the function $g(x) = xf(x)$ is: | $y=3$ | |
553fe2eb-c546-4037-a5dd-fa2db2c4e7e7 | As shown in the figure, points $$D$$ and $$E$$ are on $$AB$$ and $$AC$$ respectively. $$CD$$ and $$BE$$ intersect at point $$O$$. The following four conditions are given: 1. $$\angle ADC = \angle AEB$$; 2. $$\dfrac{AD}{AC}=\dfrac{AE}{AB}$$; 3. $$\dfrac{AD}{AB}=\dfrac{AE}{AC}$$; 4. $$\dfrac{OD}{OB}=\dfrac{OE}{OC}$$. Cho... | $$3$$ | |
da8b54e1-9f86-4f27-bb93-0f9b28ad2767 | The investigator randomly surveyed 72 male and female middle school students to find out whether they prefer liberal arts or science. The results are listed in the following table (units: people) showing the relationship between gender and preference for liberal arts or science Is there a relationship between the middl... | Yes | |
f03bc1b5-632f-4bcc-8f42-105b997df8af | As shown in the figure, lines AB and CD intersect at point O, and ray OM bisects ∠AOC. If ∠BOD = 76°, then the measure of ∠BOM is ______. | 142° | |
504261cf-2281-44cd-8f6e-45613da3350e | As shown in the figure, $$AB$$, $$AC$$, and $$BD$$ are secants of circle $$\odot O$$. Points $$P$$, $$C$$, and $$D$$ are intersection points. If $$AB=5$$ and $$AC=3$$, then the length of $$BD$$ is ___. | $$2$$ | |
5faf9ccd-4d3f-448d-be9a-a51475701ca4 | As shown in the figure, in the straight square column $$ABCD - A_{1}B_{1}C_{1}D_{1}$$, $$ \angle ADC=90^{\circ}$$, and $$AA_{1}=AD=DC=1$$, $$M \in$$ plane $$ABCD$$, when $$D_{1}M \perp $$ plane $$A_{1}C_{1}D$$, $$D_{1}M=$$___. | $$\sqrt{3}$$ | |
a6748581-ddde-4ad6-b7b9-f46687c7edf1 | In the diagram, a pentagon is given (each side of equal length) with vertices sequentially numbered as 1, 2, 3, 4, 5. Starting from any vertex, follow the edges of the pentagon in the clockwise direction. The number of the vertex indicates how many edges to move, calling each such move a 'shift'. For example, if Xiao Y... | 3 | |
3bdfaba3-7066-4f28-afb3-54d5950c9160 | As shown in the figure, in the plane rectangular coordinate system xOy, the quadrilateral ABOC is a square, and the coordinates of point A are (1,1). Arc $\overset\frown{A{{A}_{1}}}$ is centered at point B, with BA as the radius; arc $\overset\frown{{{A}_{1}}{{A}_{2}}}$ is centered at point O, with OA$_{1}$ as the radi... | (6,0)
(0,-2018) | |
a50263ee-2111-4313-9f2b-d839717503db | As shown in the figure, in a 4x4 grid paper (a total of 16 small squares), each small square is a square with a side length of 1. O, A, and B are respectively the vertices of the small square. What is the circumference of the sector OAB? (Leave the result in terms of square roots and π.) | $\sqrt{2}$π+4$\sqrt{2}$ | |
78ddafd3-d726-42ba-8296-b5f488af7c24 | As shown in the figure: In the rectangular garden $ABCD$, $AB=a$, $AD=b$, there is a rectangular path $LMPQ$ and a parallelogram path $RSTK$. If $LM=RS=c$, then the area of the part of the garden that can be greened is: | $ab-bc-ac+{{c}^{2}}$ | |
3b6b5de2-aedf-4068-97b9-0f3e5d3bd9a1 | In the figure, in $$\triangle ABC$$, $$\angle BAC=90^{ \circ }$$, $$AB=AC$$, point $$D$$ is the midpoint of side $$AC$$, $$DE \perp BC$$ at point $$E$$, connect $$BD$$, then $$\tan \angle DBC=$$___. | $$\dfrac{1}{3}$$ | |
0d7a956d-a484-4f0a-bb96-35238b90fbc5 | As shown in the figure, in the plane, rotate the right triangle ABC counterclockwise around the right angle vertex C by 90°, obtaining the right triangle EFC. If AB=\(\sqrt{5}\), BC=1, then the area of the shaded part is 。 | \(\pi-1\) | |
555caf3e-4a56-400e-a72f-a969e47f2d15 | As shown in the figure, in $$\triangle ABC$$, $$AB=AC$$. Points $$D$$ and $$E$$ are on $$BC$$. To make $$\triangle ABD\cong \triangle ACE$$, only one suitable condition needs to be added: ___. (Only fill in one!) | $$BD=CE$$ (not unique) | |
8c175953-a39e-43d5-90f3-4a69571642bf | In a sampling survey of the ages of 800 volunteers in the 'Four Cities Co-Creation' activity in a certain city, a frequency distribution histogram (as shown) was obtained, but the data for the age group [25,30) was accidentally lost. According to this graph: (1) The height of the rectangle corresponding to the age grou... | (1)$$0.04$$ (2)$$440$$ | |
be3eff1b-f1a1-462a-86b8-82910e37091b | As shown in the figure, two triangles are congruent, some lengths of sides and some angles are given, then x = ______ degrees. | In triangle ABC, ∠A=65°, ∠B=55°, therefore ∠C=180°-∠A-∠B=60°. Since the two triangles are congruent and ∠A=∠A'=65°, AB=A'C'=5cm, the corresponding point of C is B', so ∠B'=∠C=60°. Therefore, fill in 60. | |
dcfdf7d9-409c-4288-9378-bbdb0e47a81d | The position of the real number $$a$$ on the number line is shown in the figure, then $$|a-\sqrt{3}|=$$___. | $$\sqrt{3}-a$$ | |
74491a62-b9b5-4d25-b6e4-1a4d7a903e3b | As shown in the figure, four identical right triangles form a large square, with a smaller square in the middle. This constitutes a 'Zhao Shuang's chord diagram'. If the area of the large square is $$169$$, and the length of the shorter leg of the right triangle is $$5$$, then the area of the smaller square (the shaded... | $$49$$ | |
fd2dc114-1320-4da0-923a-567c25625afc | The following figures are composed of circular patterns of the same size arranged according to a certain rule. In the 1st figure, there are $3$ circular patterns; in the 2nd figure, there are $6$ circular patterns; in the 3rd figure, there are $9$ circular patterns. According to this rule, the number of circular patter... | $6057$ | |
e52b82ff-8591-44c8-a549-354cf1bb668e | Given the orthographic views of the spatial solid as shown in the figure, find the surface area of the solid: ___; and the volume of the solid: ___. | $$7 \pi$$ $$\dfrac{8 \pi }{3}$$ | |
3023bfea-0cd1-49bd-9cff-96505c4606ac | As shown in the figure, use four different colors to paint the six points $$A$$, $$B$$, $$C$$, $$D$$, $$E$$, $$F$$ in the figure. Each point should be painted with one color, and the two endpoints of each line segment should be painted with different colors. The number of different coloring methods is ___. | $$264$$ | |
348b6a45-cda9-48bc-bc8b-d501dc78e0a4 | As shown in the figure, square $$ABCD$$ is folded along $$BE$$, making point $$A$$ fall on point $$A'$$ on diagonal $$BD$$. Connect $$A'C$$, then $$\angle BA'C =$$___. | $$67.5^{\circ}$$ | |
852dfdbc-5569-4981-ba76-fdc62d612812 | As shown in the figure, in triangle ABC, PH is the perpendicular bisector of AC, AH = 3, the perimeter of triangle ABP is 11, then the perimeter of triangle ABC is. | 17 | |
a004b711-6f42-46ef-a10f-6fb6ce9d7dba | As shown in the figure, in circle $$\odot O$$, $$AB$$ is a chord, and $$C$$ is a point on $$\overparen{AB}$$. If $$\angle OAB=25^\circ$$, $$\angle OCA=40^\circ$$, then the measure of $$\angle BOC$$ is ___ degrees. | $$30$$ | |
79f3fb78-8e88-4717-9b5d-f05a3c58fb04 | As shown in the figure, in $\vartriangle \text{ABC}$, point $\text{D}$ is the midpoint of $\text{BC}$, points $\text{E}$ and $\text{F}$ are on line segment $\text{AD}$ and its extended line, respectively, and $\text{DE}=\text{DF}$. Given the following conditions: 1.$\text{BE}\bot \text{EC}$; 2.$\text{BF}//\text{EC}$; 3... | 3. | |
290fa6c2-29dd-4df4-b328-ec45e8243ef0 | As shown in the figure, quadrilateral $$ABCD$$ is congruent to quadrilateral $$A'B'C'D'$$, then $$\angle A'=$$______$$\degree$$, $$\angle A=$$______$$\degree$$, $$B'C'=$$______$$, $$AD=$$______$$. | 120
70
12
6 | |
847e75d7-756d-45a5-ab61-94dc5a27b2bb | The following figure is formed by squares with side length of $$1$$ meter. Write down their perimeter and area respectively. Perimeter: ______; Area: ______. | 14 meters
10 square meters | |
336b9238-5f41-4d13-913e-ab7d951b8f94 | As shown in the figure, points D and E are on the sides AB and AC of triangle ABC, respectively. DE is parallel to BC. AD is 4 cm, BD is 2 cm, and AC is 4.5 cm. What is the length of CE? | 1.5 | |
ac8fe0ef-12cb-43a6-abb1-7872d7b26bdf | Divide a circle with a diameter of $$4$$ centimeters into several equal parts, then cut it and arrange it as shown in the figure below. The circumference of the shape formed increases by ______ centimeters compared to the original circle's circumference. | $$4$$ | |
7bbad0e4-0c0b-45e2-8fec-c513af52e095 | As shown in the figure, with $$\triangle ABC$$, circle $$\odot O$$ has its diameter as side $$BC$$, intersecting $$AB$$ and $$AC$$ at points $$D$$ and $$E$$, respectively. Connect $$OD$$ and $$OE$$. If $$\angle A=65^{\circ}$$, then $$\angle DOE=$$___$$^{\circ}$$. | $$50$$ | |
25211fc4-8cf4-46a6-bd9a-1d7c7abdee9d | Below are four statements: 1. If each term is obtained by multiplying the previous term by the same constant, then the sequence is a geometric sequence; 2. A constant sequence b, b, b, …, b is definitely a geometric sequence; 3. In a geometric sequence, if the common ratio q = 1, then all terms of this sequence are equ... | 3.4. | |
9636b48a-56ab-4bab-977f-bd0c0111663c | As shown in the figure, there are 5 sets of data: A, B, C, D, and E. Removing one set of data, the remaining 4 sets of data have a stronger linear correlation. (Please use $A, B, C, D, E$ to answer.) | D | |
fc76f547-38ae-46eb-a1bf-c78253fe7420 | As shown in the figure, it is known that: $$ E $$ is a point on the angle bisector of $$ \angle A O B $$, $$ E C \perp O B$$, $$E D \perp O A$$, and $$C $$ and $$ D $$ are the feet of the perpendiculars. Connect $$ C D$$, which intersects $$ O E $$ at point $$F$$. If $$ \angle A O B=60\degree $$, then the relationship ... | OE=4EF | |
b25c1e19-b7a8-49fb-a88e-28d126f4ec33 | As shown in the figure, in order to measure the distance between two street lamps, Xiao Ming walks at night from street lamp $$AB$$ towards street lamp $$CD$$. When he reaches point $$E$$, he finds that the shadow of the top of his head at point $$F$$ just touches the base $$A$$ of street lamp $$AB$$. When he walks for... | 25 | |
9d6e539c-487f-425f-af14-f980b7e1ba5f | Given that a + b + c = 0, abc > 0, the value of is ______. | ∵ a + b + c = 0, abc > 0, ∴ two of a, b, c are negative and one is positive. When a and b are negative and c is positive, the original expression = -1 - 1 + 1 = -1; when a and c are negative and b is positive, the original expression = -1 + 1 - 1 = -1; when b and c are negative and a is positive, the original expressio... | |
bb7e8a5d-ad49-4e0a-9787-de8be22eb1d8 | As shown in the figure, in the tetrahedron $$D-ABC$$, if $$AB=BC$$, $$AD=CD$$, point $$E$$ is the midpoint of $$AC$$, then the relationship between plane $$ADC$$ and plane $$BDE$$ is ___. | perpendicular | |
e1a56088-e94c-49cf-87f1-1569169af2e5 | Execute the pseudocode as shown in the diagram, then the output result $S$ is. | 21 |
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