Tỷ lệ thu phóng, quan hệ góc và tam giác
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| transversal/trænsˈvɜːsl/ | A line that crosses two or more other lines. |
| exterior angle/ekˈstɪərɪə ˈæŋɡl/ | The angle between one side of a polygon and the extension of the next side. |
| congruent/ˈkɒŋɡruːənt/ | Identical in shape and size. |
A map and the real town
- On a map with scale 1 : 50 000, a park measures 4 cm by 3 cm. What is its real area?
- Similar figures have the same shape; their lengths share one scale factor — but areas and volumes scale differently.
Lengths, areas, volumes
- If lengths are multiplied by $k$, areas are multiplied by $k^2$ and volumes by $k^3$.
- Park: $4 \text{ cm} \to 2 \text{ km}$ and $3 \text{ cm} \to 1.5 \text{ km}$, so the real area is $3 \text{ km}^2$, which is $50\,000^2$ times the map area of $12 \text{ cm}^2$.
A 4 cm by 3 cm park at 1 : 50 000 is 2 km by 1.5 km, an area of 3 km²
Two similar triangles have corresponding sides 4 cm and 10 cm. The smaller triangle has area 12 cm². What is the area of the larger one in cm²?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$k = 2.5$, so area $\times 6.25$: $12 \times 6.25 = 75$.
Angles from parallel lines and triangles
- With parallel lines cut by a transversal 截线: alternate angles are equal, corresponding angles are equal, co-interior angles add to $180°$.
- Triangle angles add to $180°$; an exterior angle 外角 equals the sum of the two opposite interior angles.
In triangle $ABC$, $\angle A = 52°$ and $\angle B = 71°$. Then $\angle C = 180° - 123° = 57°$, and the exterior angle at $C$ is $52° + 71° = 123°$.
Equal or supplementary?
Use parallel-line angle relationships.
Two angles of a triangle are 48° and 65°. What is the third angle in degrees?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$180 - 48 - 65 = 67$.
A triangle has interior angles 40° and 75° at $A$ and $B$. What is the exterior angle at $C$ in degrees?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The exterior angle equals the sum of the two opposite interior angles.
Similar or congruent 全等的? Enough information?
- Similar: equal angles, proportional sides (AA is enough for triangles). Congruent: same shape and size (e.g. SSS, SAS, ASA).
- Two triangles with two pairs of equal angles are similar, but not necessarily congruent — you need a pair of equal sides as well.
Two triangles with all three pairs of angles equal must be similar.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Equal angles guarantee the same shape (similar), but the triangles may be different sizes, so not necessarily congruent.
Traps
- Scaling area by $k$ instead of $k^2$.
- Assuming congruence from equal angles alone.
A model car is built at scale 1 : 20. How many times larger is the real car's volume?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$20^3 = 8000$.
Recap
- Scale factor $k$: lengths ×$k$, areas ×$k^2$, volumes ×$k^3$.
- Parallel lines give equal alternate and corresponding angles; triangle angles sum to $180°$; AA → similar, not necessarily congruent.
- transversal: A line that crosses two or more other lines.
- exterior angle: The angle between one side of a polygon and the extension of the next side.
- congruent: Identical in shape and size.