--- title: "Tangents, Radii and the Alternate Segment Theorem" description: "Use the right angle between a tangent and the radius at the point of contact, the equal tangents from an outside point, and the alternate segment theorem." canonical: https://lightmysky.com/learn/mathematics/tangents-radii-and-the-alternate-segment-theorem-mt_Y-Xf0BOb1C source: https://lightmysky.com/learn/mathematics/tangents-radii-and-the-alternate-segment-theorem-mt_Y-Xf0BOb1C.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Tangents, Radii and the Alternate Segment Theorem Use the right angle between a tangent and the radius at the point of contact, the equal tangents from an outside point, and the alternate segment theorem. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/tangents-radii-and-the-alternate-segment-theorem-mt_Y-Xf0BOb1C ## Ready when they can - Mark the right angle where a tangent meets the radius and use it - Use the two equal tangents from an external point to find a length - Apply the alternate segment theorem to find an angle in a triangle inside the circle ## Lesson: Tangents and the alternate segment A tangent touches the circle once, and the radius to that touch meets it at 90 degrees. Mark that right angle first in every tangent problem. It turns the diagram into right-angled triangles you can solve with Pythagoras. **Example.** Two tangents from one outside point match in length. With each tangent 7 cm and chord AB 5 cm, triangle PAB has perimeter 19 cm. With radius 5 cm and centre line 13 cm, the tangent is 12 cm, since 169 minus 25 is 144. **Example.** The alternate segment theorem copies an angle across. The angle between tangent and chord equals the angle in the opposite arc: 50 degrees copies to 50. Then finish with angle sums: with angles 64 and 50, the third angle is 66. **Tip.** Name the theorem each time you move an angle. Harder problems stack the right angle with an isosceles triangle from two radii, so write the equal base angles down before chasing the rest. **Recap.** Mark the tangent right angle, match tangent pairs, copy alternate segment angles, then finish with sums. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Circle Theorems: Cyclic Quadrilaterals and the Same Segment](https://lightmysky.com/learn/mathematics/circle-theorems-cyclic-quadrilaterals-and-the-same-segment-mt_11GlmkiqB1) ## Opens up - [Lines Meeting Circles: Tangents and Chords](https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv)