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c517b12e-0ac9-43ac-8864-e3aacfd1086e
Two identical pieces of paper, each divided into 7 identical rectangles, are placed as shown in the figure. The overlapping vertex is marked as A, and vertex C lies on the dividing line DE of the other piece of paper. If BC = \sqrt{28}, then the length of AB is.
7\sqrt{2}
math
33ab3aa6-b535-492f-8a9c-c356291bafa7
The four sets of related data between variables $x$ and $y$ are shown in the table: If the regression equation between $x$ and $y$ is $\widehat{y}=\widehat{b}x+12.28$, then the value of $\widehat{b}$ is.
$-0.96$
math
45a816fa-31ab-476e-91ca-e2ac080b89a1
In the right triangle $Rt\Delta ABC$, $\angle ACB=90{}^\circ $, $CD\bot AB$, with the foot of the perpendicular at $D$, $AD=1$, $DB=4$. Find the length of $CD$.
2
math
7a826359-f495-4f75-9d2b-db9d823c23e8
As shown in the figure, the center of the ellipse $C$ is the origin $O$, and its left and right foci are $F_1$ and $F_2$, respectively. The top and bottom vertices are $A_1$ and $A_2$, respectively. It is known that point $P$ is on the ellipse $C$, satisfying $|PF_1| = |F_1F_2| = 4$. Let $Q$ be the midpoint of segment $PF_1$. If $|OQ| = 1$, then $|A_1A_2| =$.
$2\sqrt{5}$
math
62d3feb6-e16c-44ca-a90d-f919c9abd3f6
As shown in the figure, the area of quadrilateral $ABCD$ is $12$, $AD//BC$, and $BC=2AD$. Point $E$ is the midpoint of $BD$. What is the area of $\Delta BEC$?
4
math
a3da7a7c-4088-429f-a92e-49aece294d85
As shown in the figure, $AB = AC = AD$, if $AD \parallel BC$, $\angle C = 70{}^\circ$, then $\angle D$ = degrees.
$35$
math
c3293592-4465-49f3-bddb-15813a391a14
The Department of Informatization and Software Services of the Ministry of Industry and Information Technology stated that it is expected that the sales volume of new energy vehicles in China will exceed 1.5 million units in 2019, given that the sales volume of new energy vehicles in China in 2017 was 777,000 units. Let the average annual growth rate of new energy vehicle sales in China from 2017 to 2019 be $x$. According to the problem, the equation can be set as:
$77.7{{(1+x)}^{2}}=150$
math
71f684eb-109f-4c7f-8521-10a985e665a4
In recent years, outbound tourism has become an increasingly popular holiday choice for more and more Chinese citizens. The number of residents in a certain community traveling abroad in 2017 using different methods is shown in the pie chart and bar chart below. How many residents traveled as part of a tour group in 2017?
24
math
579f4314-a3ad-4fa2-9a9b-4eb0c78dba48
As shown in the figure, the line $y=kx$ ($k < 0$) intersects the hyperbola $y=-\frac{2}{x}$ at points $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$. The value of $3{{x}_{1}}{{y}_{2}}-8{{y}_{2}}{{x}_{1}}$ is.
-10
math
f8a346cc-9598-49f7-8d95-7461739acae2
As shown in the figure, in rectangle $$ABCD$$, point $$E$$ is the midpoint of side $$CD$$. After folding $$\triangle ADE$$ along $$AE$$, we obtain $$\triangle AFE$$, with point $$F$$ inside rectangle $$ABCD$$. Extending $$AF$$ intersects side $$BC$$ at point $$G$$. If $$\dfrac{CG}{GB}=\dfrac{1}{k}$$, then $$\dfrac{AD}{AB}=$$___ (express in terms of $$k$$).
$$\dfrac{\sqrt{k+1}}{2}$$
math
258b36cb-29df-414e-91a1-4d7cdb6cd42d
As shown in the figure, $$\triangle ABC$$ and $$\triangle DBC$$ are two congruent isosceles triangles sharing a common side, with $$AB=AC=3\ \unit{cm}$$ and $$BC=2\ \unit{cm}$$. $$\triangle DBC$$ is translated a certain distance along the ray $$BC$$ to obtain $$\triangle D_{1}B_{1}C_{1}$$, and $$AC_{1}$$ and $$BD_{1}$$ are connected. If quadrilateral $$ABD_{1}C_{1}$$ is a rectangle, then the translation distance is ___$$\unit{cm}$$.
$$7$$
math
21e08eba-7319-4b86-9d31-e01d9f2041ed
As shown in the figure, the side length of rhombus $$ABCD$$ is $$6$$, and $$∠ABC=60^{\circ}$$, then the length of diagonal $$AC$$ is ___.
$$6$$
math
a2034467-c5ca-48fd-8e3b-a2830c00a984
The graph of the function $$y=kx+b\left ( k≠0\right ) $$ is shown in the figure. Then the solution set of the inequality $$kx+b<0$$ is ___.
$$x < 1$$
math
bd1666e0-49a2-4c0d-879c-497e900473f1
On a Chinese geographical atlas, connecting Shanghai, Hong Kong, and Taiwan forms a triangle. The distances between them are measured as shown in the figure. The direct flight distance from Taiwan to Shanghai is approximately $$\quantity{1286}{km}$$, so the flight distance from Taiwan via Hong Kong to Shanghai is approximately ___$$\ \unit{km}$$.
$$\number{3858}$$
math
315a8676-7810-4409-90e3-7ce061c5b00f
As shown in the figure, in the parallelepiped $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, $$E$$ and $$F$$ are on $$B_{1}B$$ and $$D_{1}D$$ respectively, and $$BE=\dfrac{1}{3}BB_{1}$$, $$DF=\dfrac{2}{3}DD_{1}$$. If $$\overrightarrow{EF}=x\overrightarrow{AB}+y\overrightarrow{AD}+z\overrightarrow{AA_{1}}$$, then $$x+y+z=$$___.
$$\dfrac{1}{3}$$
math
637fcedc-5cfd-4db0-be78-37298e393f99
As shown in the figure, $$\angle B=90^{\circ}$$, $$AB=6$$, $$BC=8$$. The triangle $$\triangle ABC$$ is folded along $$DE$$ so that point $$C$$ lands on point $$C'$$ on $$AB$$, and $$C'D \parallel BC$$. Then, $$CD=$$ ___.
$$\dfrac{40}{9}$$
math