Tangents, chords and the alternate segment · Higher
| English | 中文 | Pinyin |
|---|---|---|
| alternate segment theorem/ɔːlˈtɜːnət ˈseɡmənt ˈθɪərəm/ | 弦切角定理 | xián qiē jiǎo dìng lǐ |
Two paths from an outside point just touch a circular pond. Their equal lengths follow from right-triangle congruence, not from a symmetric-looking sketch.
- Two paths from an outside point just touch a circular pond. Their equal lengths follow from right-triangle congruence, not from a symmetric-looking sketch.
- This lesson studies alternate segment theorem 弦切角定理: The tangent–chord angle equals the angle subtended by that chord in the opposite segment.
Choose the mathematical structure
- A tangent is perpendicular to its contact radius. Tangents from one external point are equal. The perpendicular from the centre to a chord bisects it. The tangent–chord angle equals the angle on that chord in the alternate segment. Use equal radii, right triangles and the central-angle theorem to prove these relationships.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines alternate segment theorem?
The tangent–chord angle equals the angle subtended by that chord in the opposite segment.
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.
At contact T, the radius OT is perpendicular to the tangent: a non-perpendicular line through T would have a smaller centre-to-line distance than the radius and cut the circle twice. For tangents PA and PB, OA=OB, OP is shared and both contact angles are 90°. RHS makes OAP and OBP congruent, so PA=PB. For a centre perpendicular OM to chord AB, OA=OB and OM is shared; RHS gives AM=MB. If OA=5 and OM=3, AM=4, hence AB=8. For a chord AB with minor central angle φ, triangle OAB has base angle (180-φ)/2. The adjacent tangent–chord angle is 90-(180-φ)/2=φ/2, equal to the circumference angle in the alternate segment. Thus a tangent–chord angle of 35° gives 35° in that segment. Reflex/supplementary configurations require the matching arc and angle.
Tangents, chords and the alternate segment
A tangent is perpendicular to its contact radius
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the radius–tangent angle.
A tangent is perpendicular to the radius at contact.
Test a tempting shortcut
- A tangent–chord angle and a radius–chord angle are different. Equal tangent lengths refer to one external point. The chord is bisected by a perpendicular from the centre, not by any line that happens to cross it.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Any line through a circle centre bisects every chord it meets. This claim is false. Explain which definition or assumption it violates.
One tangent from P has length 7. Find the other tangent length.
Tangents from the same external point have equal length.
Any line through a circle centre bisects every chord it meets.
A tangent–chord angle and a radius–chord angle are different. Equal tangent lengths refer to one external point. The chord is bisected by a perpendicular from the centre, not by any line that happens to cross it.
Interpret a new situation
- AQA G10 Higher includes these theorem proofs and related results. Give the congruence criterion or angle chain explicitly and identify the relevant chord, contact point and alternate segment.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the alternate-segment angle for a 35° tangent–chord angle.
Apply the alternate segment theorem to the same chord.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.
The tangent–chord angle equals the angle subtended by that chord in the opposite segment. Choose the relationship, show the method, check its assumptions and interpret the result.