章节大纲

  • 常规

    全部折叠 展开全部

  • 连接: 以几何方式验证公式

    We can prove the formulas used in this section geometrically. 

    ::我们可以几何地证明这段所使用的公式。

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    Perfect Square Trinomial
    ::完美广场三角广场

    1. Find the area of the image below in terms of  a  and  b. 
    ::1. 以a和b查找下方图像区域。

    2. Find the area of the shape below by finding the area of each of the four rectangles and adding them. 
    ::2. 通过找到四个矩形的每个矩形的区域并加插它们来查找下方形状的区域。

    3. Set the expressions from question 1 and question 2 equal to each other. 
    ::3. 将问题1的表达式和问题2的表达式对等。

    Difference of Two Squares
    ::两平方之差

    4. Find the area of the region that is not shaded in the figure below. 
    ::4. 找出下图中未阴影的区域。

    5. If we move the rectangle on the bottom to the side, we have the figure below. Write an expression for the area of the rectangle below.  
    ::5. 如果我们将底部的矩形移到侧面,我们就会看到下图。下面的矩形区域写一个表达式。

    6. Set the expressions from question 4 and question 5 equal to each other. 
    ::6. 将问题4的表达方式和问题5的表达方式对等。

    Sum of Two Cubes
    ::两个立方体总和

    7. Find the sum of the volumes of the two cubes below. 
    ::7. 找出以下两个立方体的体积总和。

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    8.  Next, we rearrange the cubes below. 
    ::8. 接下来,我们重新安排下面的立方体。

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    Now we will find the volume in a different way. We add imaginary lines, so these two cubes look like one large prism. Find the volume of this prism.
    ::现在我们将以不同的方式找到音量。我们添加假想的线条, 所以这两个立方体看起来像一个大棱镜。 找到这个棱镜的音量 。

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    9. Subtract the imaginary portion on top. In the picture, they are prism 1 and prism 2.
    ::9. 将上方的假想部分减去,图中为棱镜1和棱镜2。

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    10. Notice that both terms have a common factor of ( a + b ) . Factor this out. Simplify what is inside the 2nd set of parentheses. Set this expression equal to the volume expression found in question 7. 
    ::10. 请注意,这两个术语有一个共同的(a+b)系数。请将此因素考虑在内。简化第二组括号内的内容,将这个表达式设为与问题7中的体积表达式相等。

    Difference of Cubes
    ::立方体差异

    11. Pictorially, the difference of cubes looks like the image below. Imagine the smaller cube is taken out of the larger cube.
    ::11. 插图上,立方体的差别看起来像下面的图像,想象小立方体是从较大的立方体中取出的。

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    Find the volume of the difference of these two cubes.
    ::查找这两个立方体的差异大小 。

    12. Now we will find the volume in a different way. We add imaginary lines so that the shape is split into three prisms. Find the volume of prism 1, prism 2, and prism 3.
    ::12. 现在,我们将以不同的方式找到音量,我们添加假想的线条,使形状分为三个棱镜,找出棱镜1、棱镜2和棱镜3的音量。

     

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    13. Add the volumes together to get the volume of the entire shape. Pull out any common factors and simplify. Set this expression equal to the volume expression found in question 11. 
    ::13. 将各卷加在一起以获得整个形状的体积。拉出任何共同因素并简化。设置此表达式等于问题11中的体积表达式。

    PLIX 
    ::PLIX

    T ry these interactives that reinforce the concepts explored in this section:
    ::尝试这些强化本节所探讨概念的交互作用 :