Hình học: thông tin đã gán nhãn, tọa độ, đường tròn và khối hình
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| not drawn to scale/nɒt drɔːn tə skeɪl/ | A note warning that lengths and angles in a figure may not be accurate. |
| cylinder/ˈsɪlɪndə/ | A solid with two equal circular ends joined by a curved surface. |
Don't trust the drawing
- A GRE figure shows two angles that look equal. The note says "Figure not drawn to scale 未按比例绘制." Are they equal?
- Only if the given information says so. GRE geometry rewards using stated facts, coordinates and formulas — not appearance.
Use what is given
- Accept: labelled lengths and angles, marks for equal sides, stated parallel lines, straight lines (180°).
- Do not assume: right angles that are not marked, equal-looking lengths, points that look like midpoints.
Given or just apparent?
Use only stated information in GRE geometry.
In a figure “not drawn to scale”, you should not assume two angles are equal just because they look equal.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Only given information and what follows from it count; appearance can mislead.
Coordinates as evidence
- Distance: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Slope: $\dfrac{y_2 - y_1}{x_2 - x_1}$; perpendicular slopes multiply to −1.
- Points $(0, 0)$, $(6, 0)$, $(6, 8)$: legs 6 and 8, hypotenuse 10 — a right triangle proven by coordinates.
A circle has centre $(2, -1)$ and passes through $(5, 3)$. Radius $= \sqrt{9 + 16} = 5$; area $= 25\pi$.
Line $L$ has slope $\tfrac{2}{3}$. What is the slope of a line perpendicular to $L$?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Negative reciprocal.
Circles and solids
- Circle: $C = 2\pi r$, $A = \pi r^2$; arc $= \tfrac{\theta}{360} \cdot 2\pi r$. Cylinder 圆柱: $V = \pi r^2 h$. Box: $V = lwh$.
- Scaling: lengths ×k → areas ×k², volumes ×k³.
A circle has centre $(1, 2)$ and passes through $(4, 6)$. What is its radius?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$\sqrt{9 + 16} = 5$.
A box is 3 by 4 by 5. What is its volume?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$3 \times 4 \times 5$.
Common traps
- Assuming a right angle or equal sides from the picture.
- Doubling the radius and expecting the area to double (it quadruples).
The radius of a cylinder is doubled; its height is unchanged. Its volume is multiplied by
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$V = \pi r^2 h$; doubling r multiplies r² by 4.
Recap
- Use only stated facts; figures are often not to scale.
- Prove relationships with coordinates; know circle and solid formulas and how they scale.
- not drawn to scale: A note warning that lengths and angles in a figure may not be accurate.
- cylinder: A solid with two equal circular ends joined by a curved surface.