図形の語彙、性質および対称性 · 上級
導入済み| English | 意味 |
|---|---|
| line of symmetry/laɪn ɒv ˈsɪmətri/ | 対称軸 |
A square tile is also a rectangle and a rhombus. A shape may satisfy more than one definition, so the name alone does not always give every property.
- A square tile is also a rectangle and a rhombus. A shape may satisfy more than one definition, so the name alone does not always give every property.
- This lesson studies line of symmetry 对称轴: A mirror line that maps a whole shape onto itself.
Choose the mathematical structure
- A point marks a position; a line extends in both directions and a segment has two endpoints. A plane is a flat two-dimensional surface. A vertex is a corner; an edge is a boundary segment where solid faces meet. A polygon is a closed plane shape made of straight sides; regular means all sides and all interior angles are equal. Points label vertices; AB names a side and angle ABC has vertex B. Parallel lines have the same direction; perpendicular lines meet at 90°. Reflection symmetry uses a mirror line; rotational symmetry counts matches during one full turn, including 360°.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
菱形は4辺が等しい。なぜ正方形とは限らないのか？
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
正方形は4辺が等しく、かつ4つの直角を必要とする。傾いた菱形は等辺を保つが、その角度は90°と異なるため、等辺だけでは不十分である。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A square has four equal sides, four right angles, four reflection axes and rotational order 4. A non-square rectangle has opposite sides equal, four right angles, two axes and order 2. A non-square rhombus has four equal sides, opposite angles equal, two diagonal axes and order 2. A parallelogram has two parallel side pairs and opposite angles equal; a general one has no reflection axis. A kite has two pairs of adjacent equal sides; a trapezium has a pair of parallel sides. An equilateral triangle has three equal sides/angles and three reflection axes; an isosceles triangle has two equal sides and equal base angles; scalene means no equal sides. Acute, right and obtuse classify triangles by their largest angle. Pentagons, hexagons, octagons and decagons have 5,6,8,10 sides. A rectangle has equal diagonals that bisect each other. A rhombus has perpendicular bisecting diagonals; a square has both properties. For a parallelogram, a diagonal splits it into triangles: alternate angles on the two parallel side pairs agree and the diagonal is shared, so ASA establishes congruence and opposite sides are equal. Opposite-angle equality also follows from parallel-line angle facts.
図形の語彙、性質および対称性
点は位置を示し、直線は両方向に伸び、線分は2つの端点を持つ。
例題の命題を分類し、それぞれの判断を正当化する単位または不変量を説明せよ。
正方形の回転対称の次数を求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
それは every 90° と一致する: 360° 内で4回一致する。
Test a tempting shortcut
- Equal-looking lengths need stated equal-length marks or a deduction. Regular does not mean equal sides alone. An axis of symmetry is a full line, not just an internal diagonal.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every rectangle has four equal sides. This claim is false. Explain which definition or assumption it violates.
正三角形の対称軸の数を求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
各頂点から向かい合う辺の中点へ引いた線は対称軸である。
すべての長方形が4つの辺を等しく持つわけではない。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
同じ長さであることを示すには、等しい長さの記号を明記するか、減点される。正多边形は単に辺が等しいことを意味するものではない。対称軸とは内部の対角線だけでなく、完全な直線である。
Interpret a new situation
- AQA G1/G4 requires conventional names, notation and derived shape properties. State the property that justifies a classification and allow overlapping classes, such as square and rectangle.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
十角形には何個の辺があるか。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
接頭辞「deca」は10の辺を意味する。
各四角形と対角線に関する性質を対応させよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
対角線の特徴で家族分類を区別する；正方形は矩形と菱形の両方の性質を引き継ぐ。
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A mirror line that maps a whole shape onto itself. Choose the relationship, show the method, check its assumptions and interpret the result.