image imagewidth (px) 40 768 | problem stringlengths 14 673 | solution stringlengths 479 5.29k |
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<image>If the area of the gray semi-circle is 100.48, the BCEFG shape is a combination of a rectangle and an equilateral triangle, the length of the CE side is 6 and the area of the BCEFG shape is 108, compute the area of the ABC right triangle. Assume $\pi=3.14$. Round computations to 2 decimal places. | The area of the gray semi-circle is 100.48 so the length of the AC diameter can be computed as $\sqrt{\frac{8 * 100.48}{\pi}} = \sqrt{\frac{803.84}{3.14}} = \sqrt{256.0} = 16$. The area of the BCEFG shape is 108 and the length of the CE side of its rectangle is 6, so $OtherSide * 6 + \frac{\sqrt{3}}{4} * 6^2 = 108$, so... | |
<image>Question: What was the gender ratio in Sichuan in 2019? | To determine the gender ratio in Sichuan in 2019, let's follow these steps:
1. **Identify the relevant data**: The chart shows the sex ratio by region, with the female population set as 100. We need to find the value corresponding to Sichuan.
2. **Locate Sichuan in the chart**: Sichuan is listed at the very botto... | |
<image>What is the measure of angle ADC if AB is the diameter of a semicircle and C, D are two points on the semicircle such that angle CAB measures 24 degrees? | ### Problem Analysis
We are given a semicircle with diameter $ AB $. Points $ C $ and $ D $ lie on the semicircle, and we know that $ \angle CAB = 24^\circ $. We are asked to find the measure of $ \angle ADC $.
### Step-by-Step Solution
1. **Understanding the Geometry:**
- Since $ AB $ is the diameter of the... | |
<image>If the area of the ABC sector is 127.17, the BCDE shape is a combination of a rectangle and a semi-circle, the perimeter of the BCDE shape is 44 and the area of the CGHD rectangle is 126, compute the degree of the CBA angle. Assume $\pi=3.14$. Round computations to 2 decimal places. | The area of the CGHD rectangle is 126 and the length of its CG side is 23, so the length of the CD side is $\frac{126}{23} = 5.48$. The perimeter of the BCDE shape is 44 and the length of the CD side is 5.48, so $2 * OtherSide + 5.48 + \frac{5.48 * 3.14}{2} = 44$. So $2 * OtherSide = 44 - 5.48 - \frac{5.48 * 3.14}{2} =... | |
<image>Question: What alterations occurred to the Weekly Per Capita Wage ($) in Country 2 between 2013 and 2014?
Choices:
A. -2117.91
B. 1784.69
C. -1784.69
D. 2117.91 | To determine the alteration in the Weekly Per Capita Wage ($) in Country 2 between 2013 and 2014, we need to:
1. Identify the value of the Weekly Per Capita Wage in 2013.
2. Identify the value of the Weekly Per Capita Wage in 2014.
3. Calculate the difference between these two values.
### Step 1: Identify the v... | |
<image>As shown in the figure, in quadrilateral ABCD, DE bisects angle ADC, and AD = 6, BE = 2. The perimeter of quadrilateral ABCD is () | ### Problem Analysis:
We are given a quadrilateral $ABCD$ where $DE$ bisects $\angle ADC$, and we know the following:
- $AD = 6$
- $BE = 2$
We need to find the perimeter of quadrilateral $ABCD$.
### Step-by-Step Solution:
#### Step 1: Understand the properties of the quadrilateral
Since $DE$ bisects $\angl... | |
<image>What fraction of the shapes are squares?
Choices:
1/9
6/12
3/9
4/9 | To determine the fraction of the shapes that are squares, let's follow these steps:
### Step 1: Count the total number of shapes
The image contains a total of 9 shapes. We can count them one by one:
- Top row: 4 shapes
- Middle row: 4 shapes
- Bottom row: 1 shape
Total: \( 4 + 4 + 1 = 9 \)
### Step 2: Coun... | |
<image>如图,△ADB∽△ABC,若∠A=75°,∠D=45°,则∠CBD的度数是()
Choices:
A. 75°
B. 60°
C. 45°
D. 15° | To solve the problem, we need to use the properties of similar triangles and the angle relationships in triangles. Let's go through the solution step by step.
### Step 1: Understand the given information
- We are given that △ADB ∼ △ABC.
- ∠A = 75°.
- ∠D = 45°.
- We need to find the measure of ∠CBD.
### Step 2... | |
<image>Question: Which month is the wettest on average in Cairo?
Choices:
A. March
B. January
C. September | To determine which month is the wettest on average in Cairo based on the given bar chart, let's follow these steps:
1. **Examine the chart**: The chart shows the average precipitation in Cairo, Egypt, for each month of the year. The y-axis represents precipitation in millimeters (mm), and the x-axis represents the m... | |
<image>如图,AB是⊙O的弦,AC是⊙O切线,A为切点,BC经过圆心．若∠B=20°,则∠C的大小等于()
Choices:
A. 20°
B. 25°
C. 40°
D. 50° | To solve the problem, we need to use some properties of circles, including the fact that the radius is perpendicular to the tangent at the point of tangency and the properties of angles in a triangle.
### Step-by-Step Solution:
1. **Identify Key Elements:**
- $ AB $ is a chord of the circle.
- $ AC $ is a... | |
<image>Quadrilateral M N O P is a rhombus. If P R = 12, find R N. | To solve the problem, we need to use the properties of a rhombus and the given information. Let's go through the steps carefully.
### Step 1: Understand the properties of a rhombus
A rhombus has the following properties:
1. All four sides are equal in length.
2. The diagonals bisect each other at right angles (90... | |
<image>如图，矩形ABCD中，点G是AD的中点，GE⊥BG交CD于点E，CB＝CE，连接CG交BE于点F，则∠ECF的度数为（）
Choices:
A. 30°
B. 22.5°
C. 25°
D. 15° | ### 题目分析
我们需要求解矩形 $ABCD$ 中的角 $\angle ECF$ 的度数。已知条件如下：
1. 矩形 $ABCD$，其中 $AD \parallel BC$ 且 $AB \parallel CD$。
2. 点 $G$ 是 $AD$ 的中点，即 $AG = GD$。
3. $GE \perp BG$，即 $GE$ 与 $BG$ 垂直。
4. $CB = CE$，即 $E$ 是 $CD$ 上的一点，且 $CB = CE$。
5. $CG$ 与 $BE$ 交于点 $F$。
我们需要求 $\angle ECF$ 的度数。
---
### 解题步骤
#### **第一步：理解几何关系**
... | |
<image>If the ABCD shape is a combination of a rectangle and a semi-circle and the perimeter of the ABCD shape is 72, compute the length of the AB side of the ABCD shape. Assume $\pi=3.14$. Round computations to 2 decimal places. | The length of the BF side in the BCF triangle is $8$ and its opposite angle has a degree of $35$ so the length of the BC side equals $\frac{8}{tan(35)} = \frac{8}{0.7} = 11.43$. The perimeter of the ABCD shape is 72 and the length of the BC side is 11.43, so $2 * OtherSide + 11.43 + \frac{11.43 * 3.14}{2} = 72$. So $2 ... | |
<image>If the perimeter of the ABCD parallelogram is 78, the ABEF shape is a rectangle where a semi-circle has been removed from one side of it, the perimeter of the ABEF shape is 94, the length of the BH side is 11 and the length of the EH side is 14, compute the length of the AD side of the ABCD parallelogram. Assume... | The lengths of the BH and EH sides of the BHE triangle are 11 and 14, so the length of the hypotenuse (the BE side) is $\sqrt{11^2 + 14^2} = \sqrt{121 + 196} = \sqrt{317} = 17.8$. The diameter of the semi-circle in the ABEF shape is equal to the side of the rectangle with length 17.8 so the shape has two sides with equ... | |
<image>Find x. (Hint: Draw an auxiliary line). | To solve for \( x \) in the given problem, we will use the properties of parallel lines and the angles formed by a transversal. Here's the step-by-step reasoning:
### Step 1: Identify the given information
- The two horizontal lines are parallel.
- There is a transversal intersecting these parallel lines.
- The a... | |
<image>如图，直线a∥b，直线l与a、b分别交于点A、B，过点A作AC⊥b于点C．若∠1＝50°，则∠2的度数为（）
Choices:
A. 25°
B. 40°
C. 50°
D. 130° | To solve the problem, we need to analyze the given geometric configuration step by step. Let's break it down:
### Step 1: Understand the given information
- We have two parallel lines, \( a \) and \( b \).
- A transversal line \( l \) intersects \( a \) at point \( A \) and \( b \) at point \( B \).
- A line \( A... | |
<image>If the cyan shape is a combination of a rectangle and an equilateral triangle and the perimeter of the cyan shape is 66, compute the length of the side of the cyan shape marked with question mark. Round computations to 2 decimal places. | The degrees of two of the angles of the pink triangle are 45 and 35, so the degree of the angle marked with "$\theta$" $= 180 - 45 - 35 = 100$. For the pink triangle the length of one of the sides is 18 and its opposite angle is 100 so the ratio is $\frac{18}{sin(100)} = \frac{18}{0.98} = 18.37$. The degree of the angl... | |
<image>What holds all the other cell parts?
Choices:
Cytoplasm
nucleus
chloroplast
vacuole | To determine which part of the cell holds all the other cell parts, let's analyze the function and structure of each option:
1. **Cytoplasm**:
- The cytoplasm is the gel-like substance that fills the cell and surrounds the organelles.
- It provides a medium for the organelles and other cellular components to... | |
<image>If the arc length of the ABC sector is 17.99, the BCDE shape is a rectangle where a semi-circle has been removed from one side of it, the area of the BCDE shape is 96, the CDGHI shape is a combination of a rectangle and an equilateral triangle and the perimeter of the CDGHI shape is 36, compute the degree of the... | The side of the equilateral triangle in the CDGHI shape is equal to the side of the rectangle with length 6 so the shape has two rectangle sides with equal but unknown lengths, one rectangle side with length 6, and two triangle sides with length 6. The perimeter of the CDGHI shape is 36 so $2 * UnknownSide + 3 * 6 = 36... | |
<image>If the area of the yellow sector is 25.12, the area of the brown parallelogram is 84, the green shape is a combination of a rectangle and a semi-circle, the perimeter of the green shape is 50 and the angle $\omega$ is vertical to $\alpha$, compute the length of the side of the yellow sector marked with question ... | The perimeter of the green shape is 50 and the length of one side is 6, so $2 * OtherSide + 6 + \frac{6 * 3.14}{2} = 50$. So $2 * OtherSide = 50 - 6 - \frac{6 * 3.14}{2} = 50 - 6 - \frac{18.84}{2} = 50 - 6 - 9.42 = 34.58$. Therefore, the length of the side marked with letter "$a$" is $\frac{34.58}{2} = 17.29$. The leng... | |
<image>Find y.
Choices:
\frac { 15 } { 2 }
\frac { 15 } { 2 } \sqrt { 2 }
\frac { 15 } { 2 } \sqrt { 3 }
15 | To find the value of \( y \) in the given right triangle, we will use trigonometric relationships. Let's go through the solution step by step.
### Step 1: Identify the given information
- The hypotenuse of the triangle is \( 15 \).
- One of the acute angles is \( 30^\circ \).
- The side opposite the \( 30^\circ \... | |
<image>Trapezoid $ABCD$ has an area of 750 square meters. Find the height of $ABCD$.
Choices:
20
25
30
35 | Let's analyze the problem step by step.
### Step 1: Understanding the given information
- The figure shows a triangle $ \triangle ABC $ with sides $ AB = 9 $, $ AC = 9 $, and $ \angle BAC = 60^\circ $.
- The problem mentions a trapezoid $ ABCD $ with an area of 750 square meters. However, the figure only shows $ \... | |
<image>Question: In the image, can you identify the specific location with the highest altitude?
Choices:
A. Point B
B. Point A
C. Point D
D. Point C | To determine the location with the highest altitude in the image, we need to analyze the color gradient and the corresponding altitude values on the color bar:
1. **Color Bar Analysis**:
- The color bar on the right indicates that yellow represents the highest altitude, which is around 6.90 km.
- Darker shad... | |
<image>In the given figure, triangle ABC is similar to triangle BDC, with AC measuring 4 units and CD measuring 2 units. What is the length of BC? | ### Problem Analysis
We are given that triangle $ \triangle ABC $ is similar to triangle $ \triangle BDC $. This means the corresponding angles of the two triangles are equal, and the corresponding sides are proportional. We are also given the following information:
- $ AC = 4 $ units
- $ CD = 2 $ units
- We need t... | |
<image>In the diagram of the food web shown, the shrew consumes which of these directly?
Choices:
great egret
mosquito
earthworm
leopard frog | To determine which organism the shrew consumes directly in the given food web, we need to analyze the arrows pointing to the shrew. In a food web, arrows indicate the direction of energy flow, with the arrow pointing from the prey to the predator.
### Step-by-Step Reasoning:
1. **Identify the shrew in the food we... | |
<image>How many marbles are there? Estimate.
Choices:
about 80
about 40 | To estimate the number of marbles in the image, let's follow these steps:
1. **Visual Grouping**: The marbles appear to be roughly arranged in a grid-like pattern. We can estimate the number of rows and columns.
2. **Counting Rows and Columns**:
- There seem to be approximately 6 rows.
- There seem to be ... | |
<image>If the area of the ABC sector is 39.25, the degree of the EDB angle is $3x + 5$, the degree of the BED angle is $2x + 40$, the degree of the EBD angle is $3x + 95$ and the angle EBD is vertical to CBA, compute the length of the BC side of the ABC sector. Assume $\pi=3.14$. Round computations to 2 decimal places ... | The three degrees of the BDE triangle are $3x + 5$, $2x + 40$ and $3x + 95$. Therefore, $3x + 5 + 2x + 40 + 3x + 95 = 180$, so $8x + 140 = 180$, so $8x = 40$, so $x = \frac{40}{8} = 5$. The degree of the EBD angle equals $3x + 95 = 3 * 5 + 95 = 110$. The angle CBA is vertical to the angle EBD so the degree of the CBA a... | |
<image>If the yellow shape is a combination of a rectangle and a semi-circle, the perimeter of the yellow shape is 102, the perimeter of the blue parallelogram is 92, the gray shape is a rectangle where a semi-circle has been removed from one side of it and the perimeter of the gray shape is 90, compute the length of t... | The diameter of the semi-circle in the gray shape is equal to the side of the rectangle with length 16 so the shape has two sides with equal but unknown lengths, one side with length 16, and one semi-circle arc with diameter 16. So the perimeter is $2 * UnknownSide + 16 + \frac{16 * \pi}{2}$. So $2 * UnknownSide + 16 +... | |
As shown in the figure, suppose there are 4 routes from location A to location B, and 5 routes from location B to location C. Then, the total number of different routes from location A to location B, then from location B to location C, and back from location C to location A via location B, is ____.
<image> | To solve the problem, we need to determine the total number of different routes from location A to location B, then from location B to location C, and finally back from location C to location A via location B. Let's break this down step by step.
### Step 1: Analyze the routes from A to B
We are given that there are... | |
<image>Question: During this time, thermal energy was transferred from () to ().
Choices:
A. each aquarium . . . the surroundings
B. the surroundings . . . each aquarium | To answer this question, let's analyze the information provided in the table step by step:
1. **Initial Temperatures**: Both the covered aquarium and the uncovered aquarium start at the same initial temperature of 21.7°C.
2. **Final Temperatures**: After some time, both aquariums reach a final temperature of 16.2°C... | |
<image>如图,▱ABCD的周长为16cm,AC与BD相交于点O,OE⊥AC交AD于E,则△DCE的周长为()
Choices:
A. 4cm
B. 6cm
C. 8cm
D. 10cm | To solve the problem, we need to analyze the given information and use properties of parallelograms and the perpendicularity condition.
### Step-by-Step Reasoning:
1. **Properties of the Parallelogram:**
- In a parallelogram, opposite sides are equal. Therefore, if the perimeter of parallelogram $ABCD$ is $16... | |
<image>What connects the carina to the vocal fold?
Choices:
adenoids
tonsil
tongue
trachea | To determine what connects the carina to the vocal fold, let's analyze the diagram step by step:
1. **Identify the Carina**: The carina is the point where the trachea splits into the left and right main bronchi. It is clearly labeled in the diagram.
2. **Identify the Vocal Fold**: The vocal fold is located in the... | |
<image>If the arc length of the pink sector is 20.56 and the area of the brown parallelogram is 48, compute the degree of the angle marked with question mark. Assume $\pi=3.14$. Round computations to 2 decimal places. | The length of one of the sides of the brown parallelogram is 9, the area is 48 and the angle is 45. So, the sine of the angle is $\sin(45) = 0.71$, so the length of the side marked with "$a$" is $\frac{48}{9 * 0.71} = \frac{48}{6.39} = 7.51$. The radius of the pink sector is 7.51 and the arc length is 20.56. So the ang... | |
<image>If PR || KL, KN = 9, LN = 16, PM = 2 KP, find ML
Choices:
11
18
20
25 | ### Step-by-Step Solution:
#### **Step 1: Understand the given information and the diagram**
- $ PR \parallel KL $: Line $ PR $ is parallel to line $ KL $.
- $ KN = 9 $: The length of segment $ KN $ is 9.
- $ LN = 16 $: The length of segment $ LN $ is 16.
- $ PM = 2 \cdot KP $: The length of segment $ PM $ is tw... | |
<image>Does Web Green have the minimum area under the curve? | To determine whether Web Green has the minimum area under the curve, let's analyze the data step by step:
1. **Identify the Data Series:**
- The plot has three data series: Dark Gray, Mint, and Web Green.
- Each series is represented by a different color: Dark Gray (gray dots), Mint (green dots), and Web Gre... | |
Calculate the area of the trapezoid in the figure. The correct formula is (__)<image>
Options:
(A) (4+6)×7÷2
(B) (5+7)×4÷2
(C) (5+7)×6÷2 | To calculate the area of the trapezoid in the given figure, we need to use the formula for the area of a trapezoid:
\[
\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}
\]
Let's break down the problem step by step:
1. **Identify the bases and height:**
- The two parall... | |
<image>如图,为了测量山坡护坡石坝的坡度(坡面的铅直高度与水平宽度的比称为坡度),把一根长5m的竹竿AC斜靠在石坝旁,量出杆长1m处的D点离地面的高度DE=0.6m,又量得杆底与坝脚的距离AB=3m,则石坝的坡度为()
Choices:
A. \frac{3}{4}
B. 3
C. \frac{3}{5}
D. 4 | ### 题目分析
我们需要求山坡护坡石坝的坡度，即坡面的铅直高度与水平宽度的比。题目中给出了以下信息：
- 竹竿长度 $ AC = 5 \, \text{m} $。
- 杆长 1m 处的点 $ D $ 离地面的高度 $ DE = 0.6 \, \text{m} $。
- 杆底 $ A $ 到坝脚 $ B $ 的水平距离 $ AB = 3 \, \text{m} $。
我们需要利用这些信息，结合相似三角形的性质，求出石坝的坡度。
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### 步骤 1：确定竹竿与地面的夹角
竹竿 $ AC $ 的长度为 5m，点 $ D $ 距离杆底 $ A $ 为 1m，且 $ DE = 0.6 \, \text{m}... | |
<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. If angle BCD measures 110.0 degrees, what is the measure of angle BOD? | To solve the problem, we need to determine the measure of angle \( \angle BOD \) given that \( \angle BCD = 110^\circ \) in the inscribed quadrilateral \( ABCD \) with center \( O \).
### Step-by-Step Solution:
1. **Understanding the Inscribed Quadrilateral:**
- An inscribed quadrilateral (also known as a cyc... | |
<image>According to the given food chain, What is the main source of energy?
Choices:
grouse
hawk
bear
berries and floers | To determine the main source of energy in the given food chain, let's analyze the diagram step by step:
1. **Identify the Producers**: Producers are organisms that can produce their own food through photosynthesis. In the diagram, the producers are:
- **Berries and flowers**
- **Grasses**
- **Seeds**
... | |
<image>Which layer contains Oil as shown in the diagram below?
Choices:
A
F
G
C | To determine which layer contains oil in the given geological diagram, let's analyze the image step by step:
1. **Identify the Layers**: The diagram shows several layers labeled A, B, C, D, E, F, and G. These layers represent different rock formations.
2. **Understand the Rock Types**:
- **Layer D**: This app... | |
The <image> shows a square OABC with side length 10 cm that is rotated 90°, 180°, and 270° around point O. The shaded area is ____ cm². (Use π = 3.14) | To solve the problem, we need to determine the area of the shaded region in the given figure. Let's break it down step by step.
### Step 1: Understand the Geometry
- The square $ OABC $ has a side length of 10 cm.
- The square is rotated 90°, 180°, and 270° around point $ O $.
- The shaded region is the area cove... | |
<image> As shown in the figure, there is a rectangular grassland with a length of 16m and a width of 10m. Two paths have been built in the middle, one is parallelogram-shaped and the other is rectangular. What is the area of the grassland now in square meters? | To solve the problem, we need to determine the area of the grassland after accounting for the two paths. Let's break it down step by step.
### Step 1: Understand the dimensions and layout
- The grassland is a rectangle with a length of 16 meters and a width of 10 meters.
- There are two paths:
1. A parallelogra... | |
<image>If the area of the BDEF parallelogram is 72, the angle DBF is vertical to CBA and the adjacent angles BAC and BAG are complementary, compute the degree of the BCA angle. Round computations to 2 decimal places. | The lengths of the BF and the BD sides of the BDEF parallelogram are 6 and 18 and the area is 72 so the sine of the DBF angle is $\frac{72}{6 * 18} = 0.67$ and so the angle in degrees is $\arcsin(0.67) = 42.07$. The angle CBA is vertical to the angle DBF so the degree of the CBA angle = 42.07. The sum of the degrees of... | |
<image>Find F H. | We are tasked with finding the length of side $ FH $ in the given triangle $ \triangle FGH $. Let's reason through this step by step.
### Step 1: Analyze the given information
- The triangle is $ \triangle FGH $.
- The angles are given as follows:
- $ \angle F = 60^\circ $
- $ \angle G = 60^\circ $
- The si... | |
<image>What is the radius of the round glass mirror if the right-angled vertex O of the right-angled triangle is positioned on the circumference of the broken glass mirror, and the two right-angled sides and the arc intersect at points M and N respectively, with the measurements of OM and ON being 8.0 and 6.0 respectiv... | To determine the radius of the round glass mirror, let's analyze the problem step by step.
### Step 1: Understand the Geometry
- The right-angled triangle has its right-angled vertex \( O \) on the circumference of the circular mirror.
- The two right-angled sides of the triangle intersect the circular arc at poin... | |
<image>如图,△ABC∽△AED,∠ADE=80°,∠A=60°,则∠C等于()
Choices:
A. 40°
B. 60°
C. 80°
D. 100° | To solve the problem, we need to use the properties of similar triangles and the angle sum property of triangles. Let's go through the solution step by step.
### Step 1: Understand the given information
- We are given that \( \triangle ABC \sim \triangle AED \).
- This means the corresponding angles of \( \triangl... | |
<image>If the blue shape is a combination of a rectangle and an equilateral triangle, the perimeter of the blue shape is 102, the diagonal of the brown rectangle is 25 and the area of the red semi-circle is 100.48, compute the length of the side of the blue shape marked with question mark. Assume $\pi=3.14$. Round comp... | The area of the red semi-circle is 100.48 so the length of the diameter marked with "$b$" can be computed as $\sqrt{\frac{8 * 100.48}{\pi}} = \sqrt{\frac{803.84}{3.14}} = \sqrt{256.0} = 16$. The diagonal of the brown rectangle is 25 and the length of one of its sides is 16, so the length of the side marked with letter ... | |
<image>What number is shown? | To determine the number shown in the image, let's break it down step by step:
1. **Identify the Structure**: The image consists of small blue cubes arranged in a specific pattern.
2. **Count the Cubes in Each Row**:
- The first row has **3 cubes**.
- The second row has **2 cubes**.
- The third row has... | |
<image>If the length of the height perpendicular to the AB base in the ABC triangle is 19 and the diagonal of the AEFD square is 21, compute the area of the ABC triangle. Round computations to 2 decimal places. | The diagonal of the AEFD square is 21, so the length of the AD side is $\frac{21}{\sqrt{2}} = \frac{21}{1.41} = 14.89$. The lengths of the AD and BD sides of the ABD triangle are 14.89 and 11, so the length of the hypotenuse (the AB side) is $\sqrt{14.89^2 + 11^2} = \sqrt{221.71 + 121} = \sqrt{342.71} = 18.51$. For the... | |
<image>In the given diagram of rectangle ABCD, if angle BOC measures 120° and AB is equal to 4, what is the length of AC? | ### Problem Analysis
We are given a rectangle $ABCD$ with the following information:
1. $\angle BOC = 120^\circ$.
2. $AB = 4$.
3. We need to find the length of the diagonal $AC$.
#### Step 1: Understanding the Geometry of the Rectangle
- In a rectangle, the diagonals are equal in length and bisect each other. T... | |
<image>如图，在△ABC中，AB＝8，D，E分别是边AC和AB上的点，且∠AED＝∠C，若AD2AC＝26，则AE的长为（）
Choices:
A. \frac{4}{13}
B. 3
C. \frac{13}{4}
D. 4 | ### 题目分析
我们需要求线段 $ AE $ 的长度。题目中给出了以下信息：
1. $ \triangle ABC $ 是一个三角形。
2. $ D $ 和 $ E $ 分别是边 $ AC $ 和 $ AB $ 上的点。
3. $ \angle AED = \angle C $。
4. $ \frac{AD}{AC} = \frac{2}{6} = \frac{1}{3} $。
5. $ AB = 8 $。
根据这些信息，我们可以利用相似三角形的性质来解决问题。
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### 解题步骤
#### 第一步：利用角度关系判断相似三角形
已知 $ \angle AED = \angle C $，并... | |
<image>Question: Which country is highlighted?
Choices:
A. Tuvalu
B. the Marshall Islands
C. Vanuatu
D. Nauru | To determine which country is highlighted in the green box on the map, let's analyze the geographical location and context:
1. **Location**: The green box is located in the Pacific Ocean, northeast of Australia and southeast of the Philippines.
2. **Neighboring Countries/Regions**: The highlighted area is near Papu... | |
<image>Question: Think about the magnetic force between the magnets in each pair. Which of the following statements is true?
Choices:
A. The magnitude of the magnetic force is smaller in Pair 2.
B. The magnitude of the magnetic force is the same in both pairs.
C. The magnitude of the magnetic force is smaller in Pa... | To determine the magnitude of the magnetic force between the magnets in each pair, let's analyze the given information step by step:
1. **Understanding Magnetic Force**: The magnetic force between two magnets depends on the strength of the magnets and the distance between them. The force is inversely proportional to... | |
<image>Fold a rectangular paper ABDC along BE, causing points C and D to coincide with points C1 and D1, respectively. If the measure of angle C1BA is 56°, what is the measure of angle ABE? | To solve the problem, we need to analyze the geometric transformations and angles involved when the rectangular paper is folded along the line $ BE $.
### Step 1: Understand the Folding Process
When the paper is folded along $ BE $, the points $ C $ and $ D $ coincide with $ C_1 $ and $ D_1 $, respectively. This me... | |
<image>Find x.
Choices:
3 \sqrt { 6 }
9
6 \sqrt { 3 }
18 | To solve for \( x \) in the given circle problem, we need to use the Inscribed Angle Theorem. The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of the arc it intercepts.
### Step 1: Identify the inscribed angles and their intercepted arcs
- The inscribed angle \( \angle Q... | |
<image>If the CEFGH shape is a rectangle where an equilateral triangle has been removed from one side of it, the area of the CEFGH shape is 96 and the angle BCA is vertical to ECD, compute the degree of the CBA angle. Round computations to 2 decimal places. | The area of the CEFGH shape is 96 and the length of the EF side is 12, so $OtherSide * 12 - \frac{\sqrt{3}}{4} * 12^2 = 96$, so $OtherSide * 12 = 96 + \frac{\sqrt{3}}{4} * 12^2 = 96 + \frac{1.73}{4} * 144 = 96 + 0.43 * 144 = 96 + 61.92 = 157.92$. Therefore, the length of the CE side is $\frac{157.92}{12} = 13.16$. The ... | |
<image>过正方形ABCD的顶点作直线l,分别过A、C作L的垂线,垂足为E、F,若AE=3,CF=1,则AB=()
Choices:
A. 1
B. 2
C. √{10}
D. 4 | ### 题目分析
我们需要求正方形 $ABCD$ 的边长 $AB$。已知：
- 直线 $l$ 过正方形的某一点。
- 过点 $A$ 作 $l$ 的垂线，垂足为 $E$，且 $AE = 3$。
- 过点 $C$ 作 $l$ 的垂线，垂足为 $F$，且 $CF = 1$。
我们需要利用几何性质和勾股定理来解决问题。
---
### 步骤 1：设正方形的边长为 $x$
设正方形 $ABCD$ 的边长为 $x$。则 $AB = BC = CD = DA = x$。
---
### 步骤 2：利用相似三角形
由于 $AE \perp l$ 和 $CF \perp l$，直线 $AE$ 和 $CF$ 都... | |
<image>Question: What is the name of the colony shown?
Choices:
A. New Hampshire
B. Massachusetts
C. Connecticut
D. Rhode Island | To determine the name of the colony shown in dark green on the map, let's analyze the geographical context and the shape of the highlighted area:
1. **Geographical Context**: The map represents the original 13 colonies of the United States. The dark green area is located in the northeastern part of the map, which co... | |
<image>In the given figure, circle O has diameter AB and AC is a tangent. OC intersects circle O at point D and BD is connected. If angle C measures 42.0 degrees, what is the degree measure of angle ABD? | To solve for the degree measure of angle $ \angle ABD $, let's analyze the given information and use geometric properties step by step.
### Step 1: Understand the given information
- Circle $ O $ has diameter $ AB $.
- $ AC $ is a tangent to the circle at point $ A $.
- $ OC $ intersects the circle at point $ D $... | |
<image>Is Dark Gray the smoothest? | To determine whether Dark Gray is the smoothest line in the scatter plot, let's analyze the data step by step:
### Step 1: Understand the Data
- The scatter plot shows multiple data series, each represented by a different color.
- The series are labeled in the legend as Dark Violet, Medium Mint, Dark Gray, Black, ... | |
<image>Find \angle 2. | To find the measure of angle 2, we need to use the properties of parallel lines and the angles formed by a transversal. Let's go through the problem step by step.
### Step 1: Identify the given information
- We have two parallel lines (indicated by the arrow marks).
- There is a transversal intersecting these para... | |
<image>If the brown shape is a rectangle where an equilateral triangle has been removed from one side of it, the length of the height of the removed equilateral triangle of the brown shape is 8 and the area of the red semi-circle is 127.17, compute the perimeter of the brown shape. Assume $\pi=3.14$. Round computations... | The area of the red semi-circle is 127.17 so the length of the diameter marked with "$a$" can be computed as $\sqrt{\frac{8 * 127.17}{\pi}} = \sqrt{\frac{1017.36}{3.14}} = \sqrt{324.0} = 18$. For the brown shape, the length of one side of the rectangle is 18 and its other side can be computed based on the height of the... | |
As shown in the <image>, $$AB=AC$$, $$BD=BC$$. If $$∠A=40^{\circ}$$, then the degree measure of $$∠ABD$$ is $$($$__$$)$$
Options:
(A) $$20^{\circ}$$
(B) $$30^{\circ}$$
(C) $$35^{\circ}$$
(D) $$40^{\circ}$$ | We are given a triangle $ \triangle ABC $ with the following properties:
1. $ AB = AC $, so $ \triangle ABC $ is isosceles with $ \angle ABC = \angle ACB $.
2. $ BD = BC $, so $ \triangle BDC $ is isosceles with $ \angle BDC = \angle BCD $.
3. $ \angle A = 40^\circ $.
We need to find the measure of $ \angle ABD... | |
<image>Question: Which country is highlighted?
Choices:
A. Fiji
B. the Federated States of Micronesia
C. Papua New Guinea
D. New Zealand | To determine which country is highlighted in the green box on the map, let's analyze the geographical context and the location of the highlighted area:
1. **Location Analysis**:
- The highlighted area is located in the Pacific Ocean.
- It is situated northeast of Australia and to the east of Papua New Guinea... | |
<image>As shown in the figure, in circle O, AB is the diameter, AC is the chord, the tangent line passing point C and the extended line of AB intersect at point D, if angle A = 25.0, then the size of angle D is () | To solve for the size of angle $ D $, we will analyze the given geometric configuration step by step. Here is the reasoning:
### Step 1: Understand the given information
- $ AB $ is the diameter of the circle.
- $ AC $ is a chord of the circle.
- $ CD $ is a tangent to the circle at point $ C $.
- $ \angle A = 2... | |
<image>Is Orange Red the maximum? | To determine if "Orange Red" is the maximum value in the bar chart, let's follow these steps:
1. **Identify the Bars**: The chart has six bars, each corresponding to a different color: Orange Red, Blue, Coral, Light Green, Khaki, and Light Gold.
2. **Examine the Heights**:
- The height of each bar represents... | |
Given that in the figure, AB and CD are straight lines, DF intersects AB at E, and EG intersects CD at O. If ∠BEF = 124°, ∠D = 56°, and ∠DEO = 60°, what is the degree measure of ∠COE? <image> | To solve for the degree measure of ∠COE, we need to use the properties of angles formed by intersecting lines and the relationships between angles in a plane. Let's break it down step by step.
### Step 1: Analyze the given information
- **AB and CD are straight lines.**
- **DF intersects AB at E.**
- **EG interse... | |
<image>一把直尺与30°的直角三角板如图所示，∠1＝40°，则∠2＝（）
Choices:
A. 50°
B. 60°
C. 70°
D. 80° | ### 题目分析
我们需要求图中∠2的度数。根据题目描述，图中包含一把直尺和一个30°的直角三角板，且已知∠1 = 40°。我们需要利用几何中的平行线性质和三角形内角和定理来解决问题。
### 解题步骤
1. **识别平行线和角度关系**
- 直尺的两边是平行的，因此图中存在一组平行线。
- 根据平行线的性质，直尺的上下两边是平行的，我们可以利用平行线的同位角、内错角或同旁内角的性质。
2. **标记已知角度**
- 三角板中有一个30°的角，标记为∠3。
- 已知∠1 = 40°。
3. **利用平行线的性质**
- ∠1 和∠3 之间有一个内错角关系。由于直尺的上... | |
<image>Question: Among the options given in the image, which container will exhibit droplets on its inner wall first when left at room temperature?
Choices:
A. Because all temperatures are lower than the room temperature of 25°C, no containers will exhibit inner wall condensation within the given image options.
B. O... | To determine which container will exhibit droplets on its inner wall first when left at room temperature, we need to consider the concept of condensation. Condensation occurs when the temperature of the liquid inside the container is lower than the dew point of the surrounding air, causing moisture in the air to conden... | |
The positions of points a, b, and c on the number line are shown in the figure below. Which of the following expressions are correct? ( )
(1) b + c > 0, (2) a + b > a + c, (3) bc > ac, (4) ab > ac.
<image>
Options:
(A) 1 correct
(B) 2 correct
(C) 3 correct
(D) 4 correct | To determine which of the given expressions are correct, let's analyze each one step by step based on the positions of points \(a\), \(b\), and \(c\) on the number line.
### Step 1: Understand the positions of \(a\), \(b\), and \(c\)
From the number line:
- \(c\) is located between \(-2\) and \(-1\), so \(c\) is a... | |
<image> As shown in the figure, the river course is changed from curved to straight. According to ____, explain how this can shorten the navigation route. | To explain how changing the river course from curved to straight can shorten the navigation route, we need to use a fundamental geometric principle. Let's break it down step by step:
### Step 1: Understanding the Problem
The river originally follows a curved path (as shown in the figure). The goal is to change the ... | |
<image>If the blue shape is a combination of a rectangle and an equilateral triangle, the area of the blue shape is 90 and the circumference of the orange semi-circle is 15.42, compute the length of the side of the blue shape marked with question mark. Assume $\pi=3.14$. Round computations to 2 decimal places. | The circumference of the orange semi-circle is 15.42 so the diameter marked with "$a$" can be computed as $\frac{15.42}{1 + \frac{3.14}{2}} = \frac{15.42}{2.57} = 6$. The area of the blue shape is 90 and the length of one side of its rectangle is 6, so $OtherSide * 6 + \frac{\sqrt{3}}{4} * 6^2 = 90$, so $OtherSide * 6 ... | |
<image>Question: Look at the following schedule. Marie is at the boat docks at 10.30 AM How soon can she get to the art museum?
Choices:
A. 1:30 PM
B. 2:15 PM
C. 1:15 PM
D. 12:15 PM | To determine how soon Marie can get to the art museum, let's follow these steps:
1. **Identify Marie's current location and time:**
- Marie is at the boat docks at 10:30 A.M.
2. **Find the next available shuttle bus from the boat docks:**
- The shuttle bus schedule shows the following times for the boat d... | |
<image>If the ABCDE shape is a combination of a rectangle and an equilateral triangle, the length of the AB side is 15 and the area of the BFGC square is 49, compute the area of the ABCDE shape. Round computations to 2 decimal places. | The area of the BFGC square is 49, so the length of the BC side is $\sqrt{49} = 7$. To compute the area of the ABCDE shape, we can compute the area of the rectangle and add the area of the equilateral triangle. The lengths of the AB and the BC sides are 15 and 7, so the area of the rectangle is $15 * 7 = 105$. The leng... | |
<image>Question: I need the balances of trade information for Country 2 and Country 3 in 2019. Can you provide it?
Choices:
A. 2.91 x 10^12 $, -0.76 x 10^12 $
B. 0.32 x 10^12 $, -0.76 x 10^12 $
C. -0.76 x 10^12 $, 2.91 x 10^12 $
D. -0.76 x 10^12 $, 0.32 x 10^12 $ | To determine the balance of trade for Country 2 and Country 3 in 2019, we need to follow these steps:
### Step 1: Understand the Balance of Trade
The balance of trade is calculated as:
\[
\text{Balance of Trade} = \text{Export Value} - \text{Import Value}
\]
### Step 2: Extract Data for Country 2 in 2019
- *... | |
<image>Question: When placed at room temperature, which container from the options in the image will be the first to show droplets on its inner wall?
Choices:
A. Since all temperatures are lower than room temperature (25°C), there will be no dew forming on any container's inner wall within the given image options.
B... | To determine which container will be the first to show droplets on its inner wall when placed at room temperature, we need to consider the temperature of the liquid inside each container and the concept of condensation. Condensation occurs when the temperature of a surface is lower than the dew point of the surrounding... | |
<image>What process is labeled B in the diagram?
Choices:
Sedimentation
Weathering
None of the above
Melting | To determine the process labeled B in the diagram, let's analyze the context and the elements around it step by step:
1. **Identify the Context**: The diagram appears to represent a volcanic system, with various processes and features labeled. The label B is associated with a process occurring above the surface of t... | |
<image>How many dots are there?
Choices:
61
67
57 | To determine the total number of dots in the image, let's break it down step by step:
1. **Count the number of dots in each row:**
- Each row contains 10 dots.
2. **Count the number of complete rows:**
- There are 6 complete rows.
3. **Calculate the total number of dots in the complete rows:**
- \(... | |
Given the function \( f(x) = <image> \), the function is:
(A) an even function and monotonically increasing
(B) an even function and monotonically decreasing
(C) an odd function and monotonically increasing
(D) an odd function and monotonically decreasing
Options: | We are given the piecewise function:
$$
f(x) =
\begin{cases}
1 - 2^{-x}, & x \geq 0, \\
2^x - 1, & x < 0.
\end{cases}
$$
We need to determine whether this function is even, odd, or neither, and whether it is monotonically increasing, monotonically decreasing, or neither. Let's analyze step by step.
-... | |
<image>Question: Does this passage describe the weather or the climate?
Choices:
A. climate
B. weather | To determine whether the passage describes the weather or the climate, let's analyze the image and the terms in question:
1. **Weather**: Refers to short-term atmospheric conditions, such as temperature, humidity, precipitation, wind, and cloudiness, at a specific time and place. Weather can change rapidly and is of... | |
<image>Does Tan have the minimum area under the curve? | To determine whether Tan has the minimum area under the curve, let's analyze the graph step by step:
1. **Identify the Data Series:**
- The graph has three data series: Tan, Deep Pink, and Magenta.
- Tan is represented by the light brown dots.
- Deep Pink is represented by the pink dots.
- Magenta is... | |
<image>Find AC
Choices:
7.6
7.8
9.6
11 | To solve for the length of \( AC \), we need to use the properties of similar triangles. Here's the step-by-step reasoning:
### Step 1: Identify the given information
- \( \triangle ADE \) is similar to \( \triangle ABC \) because the sides are marked as parallel (indicated by the arrows).
- \( DE \parallel BC \),... | |
<image>Which layer surrounds the mantle?
Choices:
D
C
A
B | To determine which layer surrounds the mantle in the given diagram, let's analyze the structure step by step:
1. **Identify the layers:**
- The diagram shows a series of concentric circles labeled A, B, C, and D.
- The innermost layer is labeled **C**.
- The next layer outward is labeled **B**.
- The... |
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