Similar shapes · 相似图形
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| similar/ˈsɪmɪlə/ | 相似 | xiāng sì |
| scale factor/skeɪl ˈfæktə/ | 比例因子 | bǐ lì yīn zi |
| congruent/ˈkɒŋɡruːənt/ | 全等 | quán děng |
The architect's blueprint
- An architect draws a building at 1:100 scale. Every measurement on the plan is $\dfrac{1}{100}$ of the real thing.
- The drawing and the building are similar 相似 — same shape, different size.
建筑师的蓝图
- 建筑师按1:100的比例绘制建筑物。平面图上的每个测量值都是实物的$\dfrac{1}{100}$。
- 图纸和建筑物是相似的——形状相同，大小不同。
What makes shapes similar?
- Two shapes are similar if they have the same angles and all sides are multiplied by the same scale factor 比例因子 $k$.
- All circles are similar. All equilateral triangles are similar. Not all rectangles are similar.
- Shapes that are the exact same size and shape are called congruent 全等.
A triangle with sides $3, 4, 5$ and one with sides $6, 8, 10$: scale factor $k = 2$. Same angles ($90^{\circ}$), sides doubled. They are similar.
The three angles of a triangle add up to 180 degrees
什么使图形相似？
- 如果两个图形具有相同的角且所有边都乘以相同的比例因子$k$，则它们是相似的。
- 所有圆都是相似的。所有等边三角形都是相似的。并非所有矩形都是相似的。
- 大小和形状完全相同的图形称为全等。
一个边长为$3, 4, 5$的三角形和一个边长为$6, 8, 10$的三角形：比例因子为$k = 2$。角度相同（$90^{\circ}$），边长加倍。它们是相似的。
三角形的三个角之和等于180度
Similar shapes — enlargement · 相似形状——放大
Similar shapes are the same shape but a different size. An enlargement scales every length by the same factor; angles stay the same. · 相似形状是相同的形状但不同的大小。一个放大按相同的因子缩放每个长度;角保持相同。
Similar shapes have the same angles. · 相似形状有相同的角。
Similar shapes keep all angles equal; only the side lengths scale. · 相似形状保持所有角相等;只有边长缩放。
Two similar triangles have sides 5 cm and 15 cm. What is the scale factor k? · 两个相似的三角形有边 5 cm 和 15 cm。比例因子 k 是多少?
k = 15/5 = 3.
Shapes that are exactly the same size AND shape are called . · 大小和形状完全相同的形状被称为。
Congruent shapes are identical in both size and shape (scale factor = 1). · 全等形状在大小和形状上都相同(比例因子 = 1)。
Area and volume scale differently (Extended) (Extended)
- Lengths scale by $k$, but:
Don't scale area like length. If the scale factor is $2$, the area ratio is $2^2 = 4$, not $2$. And the volume ratio is $2^3 = 8$.
面积与体积的缩放规律（进阶）
- 长度按 $k$ 缩放，但：
不要像长度那样缩放面积。 如果缩放因子是 $2$，面积比是 $2^2 = 4$，而不是 $2$。而体积比是 $2^3 = 8$。
Two similar shapes have lengths in the ratio 2:3. The area ratio is 4:b. What is b? · 两个相似的形状的长度比为 2:3。面积比是 4:b。b 是多少?
Area scales by k²: 2²:3² = 4:9, so b = 9. · 面积按 k² 缩放:2²:3² = 4:9,所以 b = 9。
If the scale factor is 4, the area ratio is k²:1. What is k²? · 如果比例因子是 4,面积比是 k²:1。k² 是多少?
4² = 16. Area scales as the square of the length scale factor. · 4² = 16。面积按长度比例因子的平方缩放。
Worked example
- Two similar solids have lengths in the ratio $2:3$.
- Area ratio $= 2^2:3^2 = 4:9$.
- Volume ratio $= 2^3:3^3 = 8:27$.
- If the smaller has volume $40$ cm³: larger $= 40 \times \dfrac{27}{8} = 135$ cm³.
示例
- 两个相似立体的长度比为 $2:3$。
- 面积比为 $= 2^2:3^2 = 4:9$。
- 体积比为 $= 2^3:3^3 = 8:27$。
- 若较小立体体积为 $40$ cm³，则较大立体为 $= 40 \times \dfrac{27}{8} = 135$ cm³。
Two similar solids have lengths 2:3; the smaller has volume 40 cm³. Find the larger volume (cm³). · 两个相似的立体的长度为 2:3;较小的体积为 40 cm³。求较大的体积(cm³)。
Volume ratio 8:27, so 40 × 27/8 = 135 cm³. · 体积比 8:27,所以 40 × 27/8 = 135 cm³。
Fun fact
- The surface area of a sphere scales as $r^2$, but its volume scales as $r^3$. That's why elephants need big ears (surface area for cooling) while mice don't — volume grows faster than surface area.
Two squares are always similar — same angles, sides scaled by a factor. Here the scale factor is $2$, so the area ratio is $4$.
趣味事实
- 球体的表面积按 $r^2$ 缩放，而其体积按 $r^3$ 缩放。这就是为什么大象需要大耳朵（利用表面积散热），而老鼠不需要——因为体积的增长速度快于表面积。
两个正方形总是相似的——角度相同，边长按比例缩放。此处缩放因子为 $2$，因此面积比为 $4$。
Find a missing corresponding length
- Corresponding sides 4 cm and 10 cm give $k=10/4=2.5$ from small to large. A 6 cm side therefore becomes $L=kl=2.5(6)=15$ cm.
- In reverse, a 20 cm large side comes from $l=L/k=20/2.5=8$ cm. Core uses length ratios; Extended also uses area and volume ratios.
求缺失的对应长度
- 对应边长分别为 4 cm 和 10 cm，得出从较小到较大的比例为 $k=10/4=2.5$。因此，一条 6 cm 的边将变为 $L=kl=2.5(6)=15$ cm。
- 反之，一条 20 cm 的大边源自 $l=L/k=20/2.5=8$ cm。核心课程使用长度比；进阶课程还涉及面积和体积比。
Similar shapes have scale factor 2.5 from small to large. Find the small side corresponding to 20 cm. · 相似图形从小到大的缩放因子为 2.5。求对应于 20 cm 的小边长度。
Reverse the enlargement: 20 / 2.5 = 8 cm. · 反向放大运算：20 ÷ 2.5 = 8 cm。
You've got it
- similar = same angles, sides $\times$ scale factor $k$
- area scales by $k^2$, volume scales by $k^3$
- lengths $2:3 \Rightarrow$ areas $4:9 \Rightarrow$ volumes $8:27$
你掌握了
- 相似 = 角度相同，边长按 $\times$ 缩放因子 $k$ 缩放
- 面积按 $k^2$ 缩放，体积按 $k^3$ 缩放
- 长度 $2:3 \Rightarrow$，面积 $4:9 \Rightarrow$，体积 $8:27$