Circles — parts and theorems · 圆——各部分与定理
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| radius/ˈreɪdɪəs/ | 半径 | bàn jìng |
| diameter/daɪˈæmɪtə/ | 直径 | zhí jìng |
| circumference/sɜːˈkʌmfrəns/ | 圆周 | yuán zhōu |
| chord/kɔːd/ | 弦 | xián |
| tangent/ˈtændʒənt/ | 切线 | qiè xiàn |
| arc/ɑːk/ | 弧 | hú |
| sector/ˈsektə/ | 扇形 | shàn xíng |
| segment/ˈseɡmənt/ | 弓形 | gōng xíng |
| cyclic quadrilateral/ˈsaɪklɪk ˌkwɒdrɪˈlætərəl/ | 圆内接四边形 | yuán nèi jiē sì biān xíng |
The wheel that changed the world
- The circle is the most perfect shape in nature — every point on its edge is the same distance from the centre.
- Ancient mathematicians discovered remarkable properties hidden in this simple shape.
改变世界的轮子
- 圆是自然界中最完美的形状——它边缘上的每个点到中心的距离都相同。
- 古代数学家发现了隐藏在这个简单形状中的非凡性质。
The parts of a circle
- Radius 半径: centre to edge. Diameter 直径: right across ($= 2r$). Circumference 圆周: the distance round.
- Chord 弦: a line joining two points on the circle. Tangent 切线: touches at exactly one point.
- Arc 弧: part of the circumference. Sector 扇形: a slice between two radii. Segment 弓形: the region cut off by a chord.
A diameter is a special chord — the longest possible one, passing through the centre.
圆的各部分
- 半径(radius):中心到边缘。直径(diameter):横穿($= 2r$)。周长(circumference):绕一圈的距离。
- 弦(chord):连接圆上两点的一条线。切线(tangent):在恰好一点接触。
- 弧(arc):周长的一部分。扇形(sector):两条半径之间的一个切片。弓形(segment):由一条弦切下的区域。
一条直径是一条特殊的弦——最长的可能的一条,通过中心。
Arc and sector · 弧与扇形
Drag the angle and radius to see the arc (part of the circumference) and the sector (pie slice) it cuts off. · 拖动角和半径,看它截出的弧(圆周的一部分)和扇形(饼形切片)。
Match each circle part to its meaning. · 把每个圆的部分匹配到它的含义。
A chord joins two points; a tangent touches once; a sector is a slice between radii. · 一条弦连接两点;一条切线接触一次;一个扇形是半径之间的一个切片。
Circle theorems (Core)
- The angle in a semicircle is $90^{\circ}$ (an angle standing on a diameter).
- The angle between a tangent and a radius is $90^{\circ}$.
The main parts of a circle; a chord joins two points, a tangent touches at just one
圆定理(核心)
- 半圆(semicircle)中的角是 $90^{\circ}$(一个站在直径上的角)。
- 切线和半径之间的角是 $90^{\circ}$。
圆的主要部分;一条弦连接两点,一条切线只接触一点
The angle in a semicircle is how many degrees? · 半圆中的角是多少度?
An angle standing on a diameter is always 90°. · 一个站在直径上的角总是 90°。
A tangent to a circle meets the radius at 90°. · 一条圆的切线与半径以 90° 相遇。
The tangent is always perpendicular to the radius at the point of contact. · 切线在接触点总是垂直于半径。
Circle theorems (Extended)
- The angle at the centre is twice the angle at the circumference (same arc).
- Angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral 圆内接四边形 add up to $180^{\circ}$.
The angle at the centre is always twice the angle at the circumference for the same arc.
Same arc only. The "angle at centre = 2 × angle at circumference" rule only works when both angles subtend the same arc. Different arcs don't follow this rule.
圆定理(扩展)
- 中心处的角是圆周处角的两倍(同弧)。
- 同一弓形中的角相等。
- 一个圆内接四边形(cyclic quadrilateral)的对角加起来为 $180^{\circ}$。
对同一弧,中心处的角总是圆周处角的两倍。
只有同一弧。 "中心处的角 = 2 × 圆周处的角"规则只在两个角都张同一弧时有效。不同的弧不遵循这个规则。
The angle at the circumference is 40°. The angle at the centre on the same arc is how many degrees? · 圆周处的角是 40°。同弧上中心处的角是多少度?
Angle at centre = 2 × angle at circumference = 2 × 40 = 80°. · 中心处的角 = 2 × 圆周处的角 = 2 × 40 = 80°。
In a cyclic quadrilateral one angle is 85°. Its opposite angle is how many degrees? · 在一个圆内接四边形中,一个角是 85°。它的对角是多少度?
Opposite angles add to 180°: 180 − 85 = 95°. · 对角加起来为 180°:180 − 85 = 95°。
Angles in the same segment are . · 同一弓形中的角是。
Any two angles subtended by the same arc in the same segment are equal. · 在同一弓形中由同一弧所张的任何两个角相等。
Worked example
- Angle at circumference $= 40^{\circ}$ → angle at centre $= 2 \times 40 = 80^{\circ}$.
- In a cyclic quadrilateral, one angle $= 85^{\circ}$ → opposite angle $= 180 - 85 = 95^{\circ}$.
A sector is a slice between two radii; a segment is cut off by a chord; an arc is part of the circumference
示例
- 圆周处的角 $= 40^{\circ}$ → 中心处的角 $= 2 \times 40 = 80^{\circ}$。
- 在一个圆内接四边形中,一个角 $= 85^{\circ}$ → 对角 $= 180 - 85 = 95^{\circ}$。
一个扇形是两条半径之间的一个切片;一个弓形由一条弦切下;一条弧是周长的一部分
The missing Extended circle facts
- Alternate segment theorem: the angle between a tangent and a chord equals the angle at the circumference in the opposite segment. If that tangent-chord angle is $52^{\circ}$, the matching circumference angle is $52^{\circ}$.
- The perpendicular bisector of a chord passes through the centre; equal chords are equally far from it. Tangents PA and PB from the same outside point P have equal lengths, so $PA=7$ cm implies $PB=7$ cm.
缺失的进阶圆几何知识
- 弦切角定理： 切线与弦之间的夹角等于其对侧弓形内的圆周角。若该切弦角为 $52^{\circ}$，则对应的圆周角为 $52^{\circ}$。
- 弦的垂直平分线经过圆心；等长的弦距离圆心相等。从同一点 P 引出的两条切线 PA 和 PB 长度相等，因此 $PA=7$ cm 意味着 $PB=7$ cm。
A tangent-chord angle is 52°. Find the angle in the alternate segment in degrees. · 弦切角为 52°。求另一侧弧所对圆周角的度数。
The alternate segment theorem makes these two angles equal. · 弦切角定理表明这两个角相等。
PA and PB are tangents from the same external point P. PA is 7 cm. Find PB in cm. · PA 和 PB 是从同一点 P 引出的两条切线。已知 PA 长为 7 cm。求 PB 的长度（单位：cm）。
Tangents from the same external point have equal lengths. · 从同一点引出的两条切线长度相等。
You've got it
- angle in a semicircle $= 90^{\circ}$; tangent meets radius at $90^{\circ}$
- angle at centre $= 2 \times$ angle at circumference (same arc)
- cyclic quadrilateral: opposite angles add to $180^{\circ}$
你掌握了
- 半圆中的角 $= 90^{\circ}$;切线与半径以 $90^{\circ}$ 相遇
- 中心处的角 $= 2 \times$ 圆周处的角(同弧)
- 圆内接四边形:对角加起来为 $180^{\circ}$