images images listlengths 1 1 | problem stringlengths 63 2.77k | answer stringlengths 1 376 |
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<image>Please focus on the left half of the given image. As shown in the figure, given the linear functions y = kx - b and y = $\frac{1}{3}$x intersect at point A(a, 1), then the solution for the equation $(k-\frac{1}{3})x = b$ is x =. | 3 | |
<image>Please focus on the left half of the given image. A person is riding a bike on a road parallel to the tram tracks. He notices that a tram passes him from behind every 6 minutes and a tram passes him from the opposite direction every 2 minutes. Assuming the speeds of the trams and the person are constant (denoted... | 3 | |
<image>Please focus on the left half of the given image. There are several (more than 20) pieces of a naturally grown traditional Chinese medicine. From these, 20 pieces are randomly selected, and their weights are measured precisely in grams. It is specified that each piece of the medicine weighing no less than 15 gra... | 14, 19 | |
<image>Please focus on the left half of the given image. As shown in the figure, the side length of rhombus ABCD is 3 cm, E is the midpoint of AB, and DE is perpendicular to AB. What is the area of rhombus ABCD in cm²? | $\frac{9\sqrt{3}}{2}$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in △ABC, ∠C=90°, AC=3, AB=5, point D is a point on side BC. If △ACD is folded along AD, point C exactly lands on point E on AB, then BD=． | 2.5 | |
<image>Please focus on the left half of the given image. May 10th is Mother's Day. On Mother's Day, Xiaoming gave his mother 5 Carnations and 5 gift boxes. According to the information in the image, how much did Xiaoming spend in total? | 440 | |
<image>Please focus on the left half of the given image. Execute the program flowchart shown in the figure. If the input value of $n$ is $4$, then the output value of $s$ is. | $7$ | |
<image>Please focus on the left half of the given image. A workshop divides 10 workers into two groups, Group A and Group B, to process a certain type of part. The number of qualified parts processed by each worker in a unit of time is shown in the stem-and-leaf plot. It is known that the average number of qualified pa... | 11 | |
<image>Please focus on the left half of the given image. Given: As shown in the figure, isosceles right triangles are constructed outward on the three sides of the right triangle ABC. If the hypotenuse AB = 3, then the area of the shaded part in the figure is. | 4.5 | |
<image>Please focus on the left half of the given image. As shown in the figure, two bamboo poles AB and AD are leaning against the edge CE. It is measured that ∠ABC = α, ∠ADC = β. The ratio of the lengths of bamboo poles AB and AD is. | $\frac{\sin \beta }{\sin \alpha }$ | |
<image>Please focus on the left half of the given image. In a physical education class, the PE teacher tested the standing long jump for 40 students in Grade 9, Class 1. The scores and the corresponding number of students are shown in the figure. What is the average score of this test? | $\frac{35}{4}$ | |
<image>Please focus on the left half of the given image. In the figure, in △ABC, DE∥BC, $\frac{DE}{BC}=\frac{2}{3}$, the area of △ADE is 8, then the area of △ABC is | 18 | |
<image>Please focus on the left half of the given image. As shown in the figure, in △ABC, BP and CP are the angle bisectors of ∠ABC and ∠ACB, respectively, and PD ∥ AB, PE ∥ AC. The perimeter of △PDE is 6 cm. Then BC = ______ cm. | 6 | |
<image>Please focus on the left half of the given image. Given the following program, the output result is $$S=$$ ___. | $$56$$ | |
<image>Please focus on the left half of the given image. To understand the situation of teachers using multimedia for teaching at a school, a simple random sampling method was used to select 20 teachers from 200 teaching staff. The number of times they used multimedia for teaching last semester was surveyed and is repr... | $$100$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, the radius of $$\odot O$$ is $$OC =\quantity{5}{cm}$$, the line $$l \perp OC $$, with the foot of the perpendicular at $$H$$, and $$l$$ intersects $$\odot O$$ at points $$A$$ and $$B$$, where $$AB=\quantity{8}{cm}$$. How many $$\unit{cm}$$... | $$2$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, take two points $$C$$ and $$D$$ on the ray $$AB$$. How many rays are there in the figure? | $$4$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in rectangle $$ABCD$$, the length of $$AB$$ is $$a$$, and the length of $$BC$$ is $$b$$, where $$\dfrac{2}{3}b < a$$. The rectangular paper is folded in the following sequence, then the length of $$C'D'$$ is ___ (expressed in terms of $$a$... | $$3a-2b$$ | |
<image>Please focus on the left half of the given image. In the $$xOy$$ plane, consider the closed figure $$D$$ formed by the two semicircular arcs $$(x-1)^{2}+y^{2}=1(x \geqslant 1)$$ and $$(x-3)^{2}+y^{2}=1(x \geqslant 3)$$, and the two lines $$y=1$$ and $$y=-1$$, as shown in the shaded area in the figure. Let the so... | $$2 \pi ^{2}+16 \pi$$ | |
<image>Please focus on the left half of the given image. We can measure the distance to the Moon using the following method: as shown in the figure, during a full moon, place a 5-fen coin (with a diameter of about 2.4 cm) at a distance of about 2.6 meters from the eye (point O) at point AB, which exactly covers the Moo... | $$3.8\times 10^{5}$$ | |
<image>Please focus on the left half of the given image. In the parallelogram $$ABCD$$, point $$E$$ is the midpoint of side $$AD$$. $$EC$$ intersects diagonal $$BD$$ at point $$F$$. Then, the ratio $$EF:FC$$ equals ___. | $$1:2$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in $$\triangle ABC$$, $$DE$$ is the perpendicular bisector of $$BC$$, with the foot of the perpendicular at $$E$$, and it intersects $$AC$$ at point $$D$$. If $$AB=6$$ and $$AC=9$$, then the perimeter of $$\triangle ABD$$ is ___. | $$15$$ | |
<image>Please focus on the left half of the given image. To investigate the basic mathematical problem-solving abilities of students in a class, a random sample of students was taken to examine their scores on a mathematics test with a full score of 100 points. The score ranges were [45,55), [55,65), [65,75), [75,85), ... | 64 | |
<image>Please focus on the left half of the given image. A marksman's score in rings, denoted as $$\xi$$, has the following distribution: Given that the expectation of $$\xi$$, $$E\xi=8.9$$, what is the value of $$y$$? ______. | 0.4 | |
<image>Please focus on the left half of the given image. During the large class break activities, students actively participate in physical exercises. Xiao Hong randomly selected a portion of students from the entire school to test their 'one-minute jump rope' performance and drew the partial frequency distribution his... | 480 | |
<image>Please focus on the left half of the given image. As shown in the figure, lines $$AB$$ and $$CD$$ intersect line $$EF$$ at points $$E$$ and $$F$$, respectively, and $$∠1=105^{\circ}$$. When $$∠2=$$___, $$AB∥CD$$. | $$75^{\circ}$$ | |
<image>Please focus on the left half of the given image. Use 6 different colors to color the 'smiley face' in the figure, with the requirement that the 'eyes' (i.e., the regions marked as $$A$$ and $$B$$ in the figure) use the same color. How many different ways are there to color it? (Answer with a number) | $$216$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, the following geometric bodies are formed by arranging small cubes with an edge length of $$1$$ on the ground according to a certain pattern. If the exposed surfaces are all painted (the bottom surface is not painted), then in the $$n$$-th... | $$(8n-4)$$ | |
<image>Please focus on the left half of the given image. The figure below shows the graphs of the functions: $$y=x^{2}-1$$, $$y=x^{2}+6x+8$$, $$y=x^{2}-6x+8$$, $$y=x^{2}-12x+35$$ in the same Cartesian coordinate system. The most likely graph for $$y=x^{2}-6x+8$$ is ___. | 3 | |
<image>Please focus on the left half of the given image. As shown in the figure, the lines $$AD∥BE$$, $$AC$$ and $$BC$$ bisect $$∠BAD$$ and $$∠ABE$$ respectively, and $$∠CAD=55^{\circ}$$. Then $$∠CBE=$$ ___. | $$35^{\circ}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, $$A$$, $$B$$, and $$C$$ are three points on a number line (unit length is $$1$$), and the numbers they correspond to are all integers. If the number corresponding to point $$B$$ is $$7$$ more than twice the number corresponding to point $$... | $$3$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in rhombus $$ABCD$$, the diagonals $$AC$$ and $$BD$$ intersect at point $$O$$, and $$AC=8$$, $$BD=6$$. Then the height $$DH$$ of rhombus $$ABCD$$ is ___. | $$4.8$$ | |
<image>Please focus on the left half of the given image. The top view of a geometric solid is a rectangle as shown in the figure, the front view (or main view) is an isosceles triangle with a base length of 8 and a height of 5, and the side view (or left view) is an isosceles triangle with a base length of 6 and a heig... | $$80$$ | |
<image>Please focus on the left half of the given image. The chart below shows the scores of Xiao Hua's five math tests. The average score of Xiao Hua's five tests is ______. | 92 | |
<image>Please focus on the left half of the given image. A student walks to school, and in 10 minutes, he covers a part of the total distance (as shown in the figure). Estimating that he cannot arrive on time by walking, he switches to a taxi to reach the school. The relationship between his journey and time is shown i... | 24 | |
<image>Please focus on the left half of the given image. As shown in the figure, it is known that $$\angle 1 = 34\degree$$, then $$\angle 2 =$$______． | 56° | |
<image>Please focus on the left half of the given image. Two right-angled triangular rulers are placed with their right-angle vertices coinciding as shown in the figure. If $$∠AOD=110^{\circ}$$, then $$∠BOC=$$ ___. | $$70^{\circ}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, a square ABCD with side length 4 is folded along the crease EF, such that point B lands on the midpoint G of side AD. The area of quadrilateral BCEF is ___. | 6 | |
<image>Please focus on the left half of the given image. As shown in the figure, a residential area has a rectangular open space that is 30m long and 24m wide. It is planned to construct two identical rectangular green spaces within it, with their total area being 480m². There are pedestrian walkways of equal width bet... | 2 | |
<image>Please focus on the left half of the given image. As shown in the figure, line $$AD \parallel BE$$, $$AC$$ and $$BC$$ bisect $$\angle BAD$$ and $$\angle ABE$$ respectively, and $$\angle CAD = 55^{°}$$. What is $$\angle CBE =$$ ___. | $$35^{\circ}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, draw the perpendicular bisector of line segment $$AB$$, intersecting $$AB$$ at point $$O$$. On this perpendicular bisector, intercept $$OC = OA$$. With point $$A$$ as the center and the length of $$CA$$ as the radius, draw an arc intersect... | $$\dfrac{\sqrt{2}}{2}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, a rectangle is cut into four pieces along the dotted lines (where $$x>y$$). These four pieces can be exactly assembled into a square. If $$y=2$$, then the value of $$x$$ is ___. | $$1+\sqrt{5}$$ | |
<image>Please focus on the left half of the given image. In the semicircle shown in the figure, $$AD$$ is the diameter, and $$AD=3$$, $$AC=2$$. The value of $$\sin B$$ is ___. | $$\dfrac{2}{3}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in rectangle $$ABCD$$, $$M$$ and $$N$$ are the midpoints of sides $$AD$$ and $$BC$$, respectively, and $$E$$ and $$F$$ are the midpoints of segments $$BM$$ and $$CM$$, respectively. If $$AB=8$$ and $$AD=12$$, then the perimeter of quadrila... | $$20$$ | |
<image>Please focus on the left half of the given image. In the figure, ▱ABCD has EF passing through the intersection point O of the diagonals. If AB = 4 cm, AD = 3 cm, and OF = 1 cm, then the perimeter of quadrilateral BCFE is ______. | 9cm | |
<image>Please focus on the left half of the given image. According to the 'Thresholds and Testing for Blood and Breath Alcohol Content for Vehicle Drivers' released by the General Administration of Quality Supervision, Inspection and Quarantine, the thresholds for blood alcohol content for drivers after drinking or bei... | $$0.09$$ | |
<image>Please focus on the left half of the given image. From a physical fitness test involving 400 participants, the scores of 50 randomly selected individuals are summarized in the following table. The estimated variance of the scores for the 400 participants is ___. | 1.16 | |
<image>Please focus on the left half of the given image. The swimming ring we use when swimming can be considered as formed by rotating the following shape ___ (fill in the number) around its axis of symmetry. | 3 | |
<image>Please focus on the left half of the given image. As shown in the figure, in the Cartesian coordinate system, △P$_{1}$OA$_{1}$, △P$_{2}$A$_{1}$A$_{2}$, △P$_{3}$A$_{2}$A$_{3}$, ... are all isosceles right triangles, with their right-angle vertices P$_{1}$(3,3), P$_{2}$, P$_{3}$, ... all lying on the line y = -$\f... | $\frac{9}{{{4}^{2017}}}$ | |
<image>Please focus on the left half of the given image. The right table gives a 'triangular number array', where each column forms an arithmetic sequence, and from the third row onwards, each row forms a geometric sequence. Let the number in the $i$th row and $j$th column be ${{a}_{ij}}(i\ge j,i,j\in {{N}^{*}})$, then... | $\frac{m}{{{2}^{n+1}}}$ | |
<image>Please focus on the left half of the given image. In the figure, in rectangle $ABCD$, $AB=1$, $BC=\sqrt{3}$. A circle is drawn with $B$ as the center and $BD$ as the radius, intersecting the extension of $BC$ at point $M$. Another circle is drawn with $D$ as the center and $CD$ as the radius, intersecting $AD$ a... | $\frac{7\pi }{12}-\frac{\sqrt{3}}{2}$ | |
<image>Please focus on the left half of the given image. In the Pascal's Triangle, replace each number with a fraction to form a fraction triangle known as the Leibniz Triangle, as shown in the figure. If an ordered pair of real numbers (m, n) represents the nth number from the left in the mth row, such as (4, 2) repre... | $\frac{1}{56}.$ | |
<image>Please focus on the left half of the given image. The relationship between the quantity of bananas sold (in kilograms) and the selling price (in yuan) at a fruit shop is shown in the following table: If the quantity of bananas sold is represented by x (kilograms) and the selling price by y (yuan), then the rela... | y=3x | |
<image>Please focus on the left half of the given image. As shown in the figure, in rectangle ABCD, DE is perpendicular to AC at E, and ∠ADE:∠EDC = 3:2. What is the measure of ∠BDE? | 18° | |
<image>Please focus on the left half of the given image. Cities A and B are 600 kilometers apart. Two cars, Car A and Car B, start from City A and drive towards City B at the same time. Car A, after reaching City B, immediately returns and meets Car B on its way back. The graph below shows the distance $y$ (km) from Ci... | 75 km/h | |
<image>Please focus on the left half of the given image. As shown in the figure, person 甲 starts from point $A$ and walks in a direction $70{}^\circ $ north of east to point $B$, while person 乙 starts from point $A$ and walks in a direction $15{}^\circ $ south of west to point $C$. The degree measure of $\angle BAC$ is... | $125{}^\circ $ | |
<image>Please focus on the left half of the given image. As shown in the figure, several squares with patterns and without patterns, all of the same size, are combined to form a regular pattern. It is known that the side length of each small square is $0.3m$. Please express the relationship between the number of square... | ${{L}_{n}}=0.3(2n+1)$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in rectangle $ABCD$, $AB=10$, $AD=6$, and $E$ is a point on $BC$. When $\Delta CDE$ is folded along $DE$, point $C$ lands on point $F$ on $AB$. Find the length of $CE$. | $\frac{10}{3}$ | |
<image>Please focus on the left half of the given image. As shown in the figure, there are four points $$A$$, $$B$$, $$C$$, and $$D$$ on the number line. The point representing the opposite number of $$2$$ is point ___. | $$A$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in $$\triangle ABC$$, $$AB=2$$, $$BC=1.5$$, $$\angle ABC=120^{\circ}$$, if $$\triangle ABC$$ is rotated around the line $$BC$$ for one full revolution, then the volume of the resulting solid of revolution is ___. | $$\dfrac{3 \pi }{2}$$ | |
<image>Please focus on the left half of the given image. Let the universal set $U=\mathbb{R}$, $A=\{x\in \mathbb{N} | 1 \le x \le 10\}$, $B=\{x\in \mathbb{R} | x^2 + x - 6 = 0\}$. Then the set represented by the shaded area in the following figure is: | $\{2\}$ | |
<image>Please focus on the left half of the given image. As shown in the figure, the lengths of the two diagonals of rhombus $$ABCD$$ are $$AC=8$$ and $$BD=6$$, respectively. Therefore, the area of rhombus $$ABCD$$ is ___. | $$24$$ | |
<image>Please focus on the left half of the given image. Below is a simple numerical operation program. When the input value of $x$ is 4, the output value is. | 2 | |
<image>Please focus on the left half of the given image. As shown in the figure, in the complex plane, the complex number corresponding to point A is z, then the complex number z = ___. | 2-i | |
<image>Please focus on the left half of the given image. As shown in the figure, $$AD$$ is the median of $$\triangle ABC$$, $$AB > AC$$, $$AB = 8\unit{cm}$$, and the difference in the perimeters of $$\triangle ABD$$ and $$\triangle ACD$$ is $$2\unit{cm}$$, then $$AC = $$___$$\unit{cm}$. | $$6$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, a line parallel to $$BC$$, $$DE$$, divides $$\triangle ABC$$ into two parts of equal area. What is the value of $$\dfrac{AD}{AB}$$? | $$\dfrac{\sqrt{2}}{2}$$ | |
<image>Please focus on the left half of the given image. A company has three factories producing the same type of electronic product. The production distribution of the three factories is shown in the figure. Now, using stratified sampling, 100 items are to be drawn from the products of the three factories for a lifesp... | 50 1015 | |
<image>Please focus on the left half of the given image. The flowchart of a program is shown in the figure. After the program runs, the output value of $$x$$ is $$31$$. Then, $$a$$ equals ______. | $$3$$ | |
<image>Please focus on the left half of the given image. A clothing mall conducted a study to understand the relationship between the monthly sales volume of sweaters $$y$$ (pieces) and the average monthly temperature $$x$$ ($$\degree{\rm C}$$). They randomly collected data on the monthly sales volume and the average t... | $$46$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, if the front view of a cone is an isosceles right triangle with a hypotenuse of length $$4$$, then the lateral surface area of the cone is ___. | $$4\sqrt{2} \pi$$ | |
<image>Please focus on the left half of the given image. In an opaque bag, there are $$n$$ balls that are identical except for their color, including $$5$$ black balls. A ball is randomly drawn from the bag, its color is noted, which is called one ball-drawing trial, and then it is put back into the bag. The bag is mix... | $$10$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in the right triangle $$\text{Rt}\triangle ABC$$, $$∠C=90^{°}$$, $$BC=3$$, $$AC=6$$, point $$D$$ is a moving point on the side $$AC$$, and point $$D$$ moves from point $$C$$ to point $$A$$. If $$CD=x$$ and the area of $$\triangle ABD$$ is ... | $$y=9 -\dfrac{3}{2}x$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, $$\triangle ABC \cong \triangle DEF$$, $$AD = 10\ \unit{cm}$$, $$BE = 6\ \unit{cm}$$, then the length of $$AE$$ is ___. | $$\quantity{2}{cm}$$ | |
<image>Please focus on the left half of the given image. Two masts on a ship, each 7.5m high, are 15m apart. A 30m long rope is tied to the tops of the masts and is stretched as shown in the figure. Assuming the rope lies in the plane containing the two masts, the distance from the point P where the rope touches the de... | 4.75 | |
<image>Please focus on the left half of the given image. As shown in the figure, in $$\triangle ABC$$, point $$D$$ is on side $$BC$$, $$AD \bot AC$$, $$\sin\angle BAC = \dfrac{2\sqrt{2}}{3}$$, $$AB = 3\sqrt{2}$$, $$AD = 3$$, then the length of $$BD$$ is ___. | $$\sqrt{3}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in the rhombus $$ABCD$$, $$AB=10$$, $$AC=12$$, then its area is ___. | $$96$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, this is a cross-section of a water pipe with a diameter of $$2m$$ laid horizontally. The width of the water surface is $$1.6m$$. The deepest point of the water in the pipe at this time is ___$$m$$. | $$0.4$$ | |
<image>Please focus on the left half of the given image. The English oral test scores (unit: points) of 10 students in groups $$A$$ and $$B$$ are as follows: Group $$A$$: $$82$$, $$84$$, $$85$$, $$89$$, $$79$$, $$80$$, $$91$$, $$89$$, $$79$$, $$74$$; Group $$B$$: $$76$$, $$90$$, $$84$$, $$86$$, $$81$$, $$87$$, $$86$$, ... | $$B$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, given that $$O$$ is the origin, point $$A\left (3,0\right )$$, $$B\left (4,4\right )$$, and $$C\left (2,1\right )$$, the coordinates of the intersection point $$P$$ of $$AC$$ and $$OB$$ are ___. | $$\left (\dfrac{3}{2},\dfrac{3}{2} \right )$$ | |
<image>Please focus on the left half of the given image. According to the program shown in the figure, the output value of $$k$$ is ___. | $$3$$ | |
<image>Please focus on the left half of the given image. Draw a square, then connect the midpoints of each side of this square to form a second square, and so on. In this way, a total of 3 squares are drawn, as shown in the figure. If a point is randomly thrown into the figure, what is the probability that the point la... | $$\dfrac{1}{4}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, given points $$A(a,0,0)$$, $$B(0,b,0)$$, $$C(0,0,c)$$, then a normal vector of plane $$ABC$$ is ___. | $$(bc,ac,ab)$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in the tetrahedron $$C-ABD$$, $$E$$ and $$F$$ are the midpoints of $$AC$$ and $$BD$$ respectively. If $$CD=2AB=4$$ and $$EF \bot AB$$, then the angle formed by $$EF$$ and $$CD$$ is ___. | $$30^{\circ}$$ | |
<image>Please focus on the left half of the given image. The figure below shows a simple numerical operation program. When the input value of $$x$$ is $$-1$$, the output value is ___. | $$-2$$ | |
<image>Please focus on the left half of the given image. A factory conducts a sampling inspection of a batch of products. Based on the net weight (unit: grams) data of the inspected products, a frequency distribution histogram is drawn as shown in the figure. It is known that the range of the product's net weight is $$... | $$60$$ | |
<image>Please focus on the left half of the given image. In the right triangle $$ABC$$, $$\angle B=90^{\circ}$$, the perpendicular bisector of $$AC$$, $$DE$$, intersects $$AC$$ at point $$D$$ and $$BC$$ at point $$E$$. If $$\angle BAE=10^{\circ}$$, then the measure of $$\angle C$$ is ___. | $$40^{\circ}$$ | |
<image>Please focus on the left half of the given image. To welcome the school arts festival, a class in the seventh grade is holding a class song collection activity, with submissions accepted from Monday to Friday. The class committee grouped the number of works submitted by the students by day and created a frequenc... | $$48$$ | |
<image>Please focus on the left half of the given image. In the stem-and-leaf plot shown, the sum of the medians of Group A and Group B is | 64 | |
<image>Please focus on the left half of the given image. As shown in the figure, $$\triangle ABC$$ is an equilateral triangle, and $$BM=CN$$. $$AM$$ and $$BN$$ intersect at point $$P$$. The measure of $$\angle APN$$ is ______ degrees. | 60 | |
<image>Please focus on the left half of the given image. As shown in the figure, extend line segment $$AB$$ to $$C$$, such that $$BC=4$$. If $$AB=8$$, then the length of line segment $$AC$$ is ___ times the length of $$BC$$. | $$3$$ | |
<image>Please focus on the left half of the given image. The graph of the function $$y=f(x)=a^{x}-\dfrac{1}{a}$$ (where $$a>0$$ and $$a \neq 1$$) could be ___. | 4 | |
<image>Please focus on the left half of the given image. As shown in the figure, given the hyperbola $$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1(a>0,b>0)$$ with the left focus at $$F_1$$, the left and right vertices at $$A$$ and $$B$$ respectively, and point $$M$$ on the hyperbola and above the $$x$$-axis, $$MF_1⊥x$$-axis. T... | $$3+2\sqrt{2}$$ | |
<image>Please focus on the left half of the given image. The figure below is a flowchart of an algorithm. Answer the following question: When the input value is $$3$$, the output result is ___. | $$8$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, regular octagons with side lengths of $$1$$, $$2$$, and $$3$$ are stacked together, with the distance between adjacent beads on the same side being $$1$$. If regular octagons with side lengths of $$4$$, $$5$$, $$6$$, $$\cdots$$, $$10$$ are... | $$341$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in $$\triangle ABC$$, it is given that $$\overrightarrow{AN}=\dfrac{1}{2}\overrightarrow{AC}$$, and $$P$$ is a point on $$BN$$. If $$\overrightarrow{AP}=m\overrightarrow{AB}+\dfrac{1}{4}\overrightarrow{AC}$$, then the value of the real num... | $$\dfrac{1}{2}$$ | |
<image>Please focus on the left half of the given image. As shown in the figure, $$\triangle ABC$$ is an equilateral triangle with a side length of $$2\sqrt{3}$$, and $$P$$ is any point on the circle centered at $$C$$ with a radius of $$1$$. Then, $$\left (\overrightarrow{AP}\cdot \overrightarrow{BP}\right )_{ \min }=$... | $$1$$ | |
<image>Please focus on the left half of the given image. The graph shown is of the inverse proportion function $y=\frac{k}{x}$ in the second quadrant. If the area of rectangle OABC in the graph is 2, then k=． | -2 | |
<image>Please focus on the left half of the given image. As shown in the figure, the light bulb $P$ is directly above the horizontal rod $AB$. The shadow of $AB$ under the light is $CD$, and $AB \parallel CD$. Given that $AB = 1.5m$, $CD = 4.5m$, and the distance from point $P$ to $CD$ is $2.7m$, what is the distance b... | $1.8$ | |
<image>Please focus on the left half of the given image. As shown in the figure, in rectangle ABCD, DC = 6 cm. There is a point E on DC. Folding triangle ADE along line AE makes point D fall exactly on side BC, and this point is labeled as F. If the area of triangle ABF is 24, then the length of CE is cm². | $\frac{8}{3}$ | |
<image>Please focus on the left half of the given image. As shown in the figure, a right-angled triangular board ABC with a ${30}^\circ$ angle is rotated around point A, so that points B, A, and C′ lie on the same straight line. The angle of rotation of the triangular board ABC is _____ degrees. | 150° |
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