Example Question - rectangles
Here are examples of questions we've helped users solve.
Solving a Geometric Problem with Hexagon and Rectangles
이 문제는 다음과 같이 풀 수 있습니다. 문제에서 주어진 다면체가 있는데, 그 중 윗면과 아랫면이 평행한 육각형이고, 나머지 면들은 모두 사각형입니다. 윗면과 아랫면의 한 변의 길이가 a로 주어지고, 사각형의 두 변의 길이가 각각 b와 c로 주어졌습니다. 육각형의 둘레를 구하려면 육각형 한 변의 길이에 변의 수(6개)를 곱해야 합니다. 육각형 둘레 = 6a 다음으로 사각형이 육각형의 변마다 하나씩 있으므로, 사각형의 세로 변의 길이의 총 합은 각 사각형 한 변의 길이 b에 사각형의 수(6개)를 곱해야 합니다. 사각형 세로 변의 총 길이 = 6b 마지막으로, 사각형의 가로 변 c도 사각형의 수와 같으므로, 가로 변의 총 길이도 구합니다. 사각형 가로 변의 총 길이 = 6c 이제 모든 변의 총 길이를 합쳐야 합니다. 전체 둘레 = 육각형 둘레 + 사각형 세로 변의 총 길이 + 사각형 가로 변의 총 길이 전체 둘레 = 6a + 6b + 6c 문제에서 요구하는 것은 "2a - b + c"의 값을 구하는 것이므로, 이 식에서 공통적인 6을 배수로 이용하여 전체 둘레를 6으로 나누어주면 원하는 값을 얻을 수 있습니다. (6a + 6b + 6c) ÷ 6 = a + b + c 따라서 2a - b + c의 값을 구하려면 식을 변형하여 다음과 같이 구합니다. 2(전체 둘레 ÷ 6) - b + c = 2(a + b + c) ÷ 6 - b + c = (2a + 2b + 2c) ÷ 6 - b + c = (2a/6 + 2b/6 + 2c/6) - b + c = (a/3 + b/3 + c/3) - b + c = a/3 - (2/3)b + (4/3)c 이 식을 통해 문제에서 요구하는 "2a - b + c" 값을 구할 수 있습니다.
Calculating Surface Area of a Triangular Prism
To find the surface area of the triangular prism shown in the image, we can sum the areas of all the individual faces of the prism. The prism has two triangular faces and three rectangular faces. We'll calculate each of these separately and then add them together: 1. The area of one triangular face: Since both triangular faces are identical, we only need to calculate the area for one and then double it. The formula for the area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). Here, the base is 5 cm and the height is 12 cm. Area of one triangle = \( \frac{1}{2} \times 5 \, \text{cm} \times 12 \, \text{cm} = 30 \, \text{cm}^2 \) Since there are two triangles, the total area for the triangular faces: 2 × 30 cm² = 60 cm² 2. The area of the three rectangular faces: We need to consider all three rectangles: the bottom, the side that is parallel to the triangular base, and the back rectangle. - The bottom rectangle has dimensions of 6 cm (length) and 5 cm (width). Area of the bottom rectangle = length × width = 6 cm × 5 cm = 30 cm² - The side rectangle (parallelogram face) has dimensions of 6 cm (base) and 12 cm (slant height). Area of the side rectangle = base × height = 6 cm × 12 cm = 72 cm² - The back rectangle has dimensions of 12 cm (length) and 5 cm (width). Area of the back rectangle = length × width = 12 cm × 5 cm = 60 cm² Adding the area of all three rectangles: 30 cm² + 72 cm² + 60 cm² = 162 cm² Finally, add the areas of the triangles and rectangles together to get the total surface area: Total surface area = area of triangles + area of rectangles Total surface area = 60 cm² (triangles) + 162 cm² (rectangles) Total surface area = 222 cm² So, the surface area of the triangular prism is 222 cm².
Calculating Area of Composite Figure Using Rectangles
To find the area of the composite figure shown in the image, we can divide it into simpler shapes (such as rectangles) and then calculate the area of each before summing them up. From the image, it appears that we can divide the figure into two rectangles: 1. The larger rectangle on the left, which has a width of 3 cm (since the entire bottom length is 8 cm, and the length to the right is 5 cm, the difference is 8 cm - 5 cm = 3 cm) and a height of 6 cm. 2. The smaller rectangle on the right, which has a width of 5 cm and a height of 2 cm (since the entire length on the left is 6 cm, and the topmost length is 4 cm, the difference is 6 cm - 4 cm = 2 cm). Let's calculate their areas: For the larger rectangle: Area = width × height = 3 cm × 6 cm = 18 cm² For the smaller rectangle: Area = width × height = 5 cm × 2 cm = 10 cm² Now, we add the areas of the two rectangles together to get the total area of the figure: Total Area = Area of larger rectangle + Area of smaller rectangle Total Area = 18 cm² + 10 cm² = 28 cm² So, the area of the figure is 28 square centimeters.
Calculating Area of Composite Figure with Rectangles
To find the area of this composite figure, which is a combination of rectangles, we can approach it by breaking down the shape into simpler parts that we can easily calculate the area for and then combine them. From the image, we can see that there is a large rectangle on the right side of the figure with the dimensions of 5 cm by 8 cm. Next to it on the left, there is an upside-down "L" shaped figure which can be broken down into two smaller rectangles - one with the dimensions of 3 cm by 3 cm and the other 2 cm by 5 cm. Let's calculate the area for each part: 1. Large rectangle: 5 cm x 8 cm = 40 cm² 2. Small square (3 cm by 3 cm): 3 cm x 3 cm = 9 cm² 3. Small rectangle (2 cm by 5 cm): 2 cm x 5 cm = 10 cm² Now let's add up the areas of all parts: Total area = Large rectangle area + Small square area + Small rectangle area Total area = 40 cm² + 9 cm² + 10 cm² Total area = 59 cm² So, the area of the figure is 59 square centimeters.
Calculating Area of L-Shaped Polygon
The image displays an L-shaped polygon with dimensions given in units. To find the area of the polygon, we would typically divide the shape into smaller, more manageable rectangles or squares, calculate the area of each, and then combine these areas. However, in this case, it's simpler to subtract the area of the missing rectangle from the area of the larger rectangle that would encompass the entire L-shape. So first, let’s find the outer rectangle’s dimensions if the L-shape was closed to form a rectangle. The width of the rectangle would be 15 units (as given), and the length would be 7 + 7 = 14 units. Now calculate the outer rectangle’s area: 15 units * 14 units = 210 square units. Next, subtract the area of the missing bottom-right rectangle, which has dimensions of 13 units by 7 units: 13 * 7 = 91 square units. Finally, subtract the area of the smaller rectangle from the area of the larger rectangle: 210 square units - 91 square units = 119 square units. So, the area of the L-shaped polygon is 119 square units.
Solving Area of Rectangles Inside Each Other
The image shows two rectangles, one inside the other. The larger rectangle has a length described as "x + 8" and a width described as "x + 5". The smaller rectangle, presumably positioned inside the larger one, has a length described as "x + 1" and a width described as "x". To solve a question involving these rectangles, we would typically be asked to express the area of one in terms of the other or find the dimensions of the rectangles given certain conditions. However, since the specific question isn't stated in the image, I'll provide a general approach. If we were to find the area of each rectangle, we would: 1. Calculate the area of the larger rectangle by multiplying its length and width: Area = (x + 8)(x + 5) 2. Calculate the area of the smaller rectangle by multiplying its length and width: Area = (x + 1)(x) Without additional information or a specific question, we cannot solve further. If you have a particular equation or condition that relates these rectangles, please provide that for further assistance.
Calculating Area of Irregular Figure Using Rectangles
To find the area of the irregular figure drawn in the image, we can split it into simpler shapes, such as rectangles, compute their areas, and then sum these areas. The figure can be split into three rectangles labeled A, B, and C in the image. Here's how to calculate each rectangle's area: - Rectangle A has a width of 4m. To find its height, we subtract the width of Rectangle C (which is 8m) from the total width of the structure (which is 12m). This gives us 12m - 8m = 4m. Therefore, the area of Rectangle A is 4m x 4m = 16m². - Rectangle B is directly beneath Rectangle A, sharing the width of 4m, and stretches across the entire height of the structure, which is 8m. Therefore, the area of Rectangle B is 4m x 8m = 32m². - Rectangle C has a width of 8m and also stretches across the entire height of the structure, 8m. Its area is 8m x 8m = 64m². Finally, we sum the areas of Rectangles A, B, and C to get the total area of the figure: Total area = Area of A + Area of B + Area of C Total area = 16m² + 32m² + 64m² Total area = 112m² So, the area of the irregular figure is 112 square meters.
Calculating Area of Irregular Figure with Rectangles
The image shows calculations made to find the area of an irregular figure that is composed of two rectangles, labeled A and B. However, there seems to be a mistake in the calculations presented in the image. Let me guide you through the correct process to find the total area. Firstly, to find the area of each part of the irregular figure, you need to multiply the width by the height of each rectangle. For rectangle A: - The width is given as 6 meters. - The height is given as 18 meters. So, the area of A will be 6 meters x 18 meters = 108 square meters. For rectangle B: - The width is not directly given; however, B is an extension of A, which makes B's full length equal to the length of A (18 meters) plus the additional length equal to the width of A (6 meters). - Thus, B's length is 18 m + 6 m = 24 meters. - The height of B is given as 18 meters. So, the area of B will be 24 meters x 18 meters = 432 square meters. Add the area of A and B to find the total area of the irregular shape: Area of A + Area of B = 108 square meters + 432 square meters = 540 square meters. The correct answer is 540 square meters for the area of the entire irregular figure.
Calculating Area of L-Shaped Figure
To find the area of the L-shaped figure, it's best to divide the shape into two rectangles and then calculate the area of each rectangle separately. From the image, we can see that the entire length of the bottom edge is 18 meters. However, for the vertical section (rectangle A), we only have the height, which is 6 meters. The width of that section is missing, but it can be calculated by subtracting the width of the horizontal section (rectangle B) which can be inferred by the height of rectangle A (since both are likely the same given the symmetry of the L-shape) from the total length of 18 meters. So, width of A = 18m - 6m = 12m Now we have the dimensions for both rectangles. Here are the calculations for their areas: Area of rectangle A (vertical section): A = length × width = 6m × 12m = 72m² Area of rectangle B (horizontal section): B = length × width = 18m × 6m = 108m² Finally, sum the two areas to find the total area: Total area = Area of A + Area of B = 72m² + 108m² = 180m² So, the area of the L-shaped figure is 180 square meters.
Understanding Triangle Similarity in Rectangles
The image depicts problem number 3, which asks to explain why triangle \(\triangle FE\)C must be similar to triangle \(\triangle ABC\) in the given rectangle \(ABCD\) with the diagonal \(AC\). From the given diagram, we can observe that rectangle \(ABCD\) consists of \(\triangle ABC\) and \(\triangle ADC\), which are congruent to each other since both are right triangles sharing the diagonal \(AC\) as the hypotenuse. By the definition of a rectangle, we know: a) The opposite sides are equal (AD = BC and AB = DC). b) All angles are right angles (90 degrees). Point \(F\) lies on \(AB\) and point \(E\) lies on \(DC\), such that \(FE \parallel AC\). Due to the parallel lines, we can deduce the following: 1. Angle \(BAC\) is congruent to angle \(FEC\) because they are corresponding angles. 2. Angle \(ABC\) is congruent to angle \(EFC\) because they are also corresponding angles. Since \(\triangle ABC\) and \(\triangle FEC\) share the same angle (\(C\)) and have two angles congruent to each other, by the Angle-Angle (AA) postulate, the two triangles are similar. To summarize, \(\triangle FEC\) is similar to \(\triangle ABC\) because they have two angles congruent by parallel lines (\(FE \parallel AC\)) creating corresponding angles, and they share the angle at \(C\).
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