الهندسة: الحقائق المسمّاة، الإحداثيات، الدوائر والأجسام الصلبة
GRE General Test الموضوع 13 2:31 سرد باللغة الإنجليزية · ترجمة مدمجة بالإنجليزية + الصينية
فصول
النص الصوتي
This unit uses given relationships rather than a diagram's apparent scale, connects coordinate distances, slopes and geometric properties, and distinguishes linear, area and volume scale factors.
本单元依据给出的关系而不是图形的视觉比例，把坐标距离、斜率与几何性质联系起来，并区分长度、面积和体积的比例因子。
Extract the labelled facts: lengths, right-angle marks, parallel lines and radii; do not measure the screen.
提取图中标注的事实：长度、直角符号、平行线和半径；不要量屏幕。
Pythagoras needs a right triangle.
勾股定理需要直角三角形。
With coordinates, use the midpoint and distance formulas; slope is delta y over delta x, and perpendicular slopes multiply to minus 1.
有坐标时用中点和距离公式；斜率是 Δy/Δx，互相垂直的直线斜率之积为 −1。
A circle has circumference 2 pi r and area pi r squared, and a sector uses its fraction of a full turn.
圆的周长是 2πr，面积是 πr²，扇形取整圈的相应比例。
Boxes use l w h and cylinders pi r squared h.
长方体体积 lwh，圆柱 πr²h。
For similar figures with length factor k, areas scale by k squared and volumes by k cubed.
长度比为 k 的相似图形，面积乘以 k²，体积乘以 k³。
Points A minus 1, 2 and B 5, 10 have delta x equals 6 and delta y equals 8.
点 A(−1, 2) 与 B(5, 10) 的 Δx = 6，Δy = 8。
Distance is 10, from a 6, 8, 10 right triangle, and the midpoint 2, 6.
由 6、8、10 直角三角形得距离 10，中点 (2, 6)。
A 60 degree sector of radius 6 is one sixth of the circle: area 6 pi, arc 2 pi.
半径 6、圆心角 60° 的扇形是整圆的六分之一：面积 6π，弧长 2π。
Use labelled facts, include required pi or units, and distinguish a perimeter from an area or a sector from its arc.
使用标注的事实，写上需要的 π 或单位，并区分周长与面积、扇形与弧长。
The cylinder's volume is pi times 3 squared times 4, 36 pi cubic centimetres.
圆柱体积是 π × 3² × 4 = 36π 立方厘米。
Its total surface is two ends of 9 pi each plus the side, 2 pi times 3 times 4: 42 pi square centimetres.
总表面积是两个底面各 9π，加上侧面 2π × 3 × 4：共 42π 平方厘米。
Enlarging every length by 3 multiplies area by 9 and volume by 27.
所有长度放大 3 倍，面积乘以 9，体积乘以 27。
A rectangle has perpendicular sides, so its diagonal makes a right triangle and the other side is 8.
长方形的邻边垂直，所以对角线构成直角三角形，另一边是 8。
Area nine times as large means length three times, so the sides are 18 and 24 and the perimeter 84.
面积是 9 倍意味着长度是 3 倍，所以边长为 18 和 24，周长 84。
An arbitrary quadrilateral would not justify Pythagoras.
任意四边形不能用勾股定理。
Check yourself.
自我检测。
The distance is 10 and the midpoint 2, 6.
距离是 10，中点 (2, 6)。
The sector has area 6 pi.
扇形面积 6π。
The cylinder's surface is 42 pi.
圆柱表面积 42π。
And an area nine times larger triples the perimeter, from 28 to 84.
面积变为 9 倍，周长变为 3 倍，从 28 到 84。