--- 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: 2026-11-12 --- > **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 line meets a circle at one point. The radius ending at that contact point is perpendicular to the tangent: they meet at 90 degrees. The converse also holds. A line through a point on the circle is tangent there if it is perpendicular to the radius ending there. The rule concerns the contact radius, not every radius. Tangent segments PA and PB from one outside point P to the same circle have equal lengths. Join centre O to A, B and P. Right triangles OAP and OBP share hypotenuse OP and have equal radius legs, so they are congruent. Thus PA = PB. To find a perimeter, add both tangents and the chord. If the two tangents total L, each is L / 2. *(drawing: With chord AB = 6 cm, the equal tangent lengths give triangle PAB a perimeter of 22 cm. This shows the calculation, not the geometry.)* For tangent PT and contact radius OT, triangle OTP is right-angled at T. OP is the hypotenuse. Pythagoras gives OP^2 = OT^2 + PT^2. Add the squared legs to find OP. Subtract a squared leg from OP^2 to find the other leg. Take the positive square root for a length. For OT = 7 cm and PT = 24 cm, OP^2 = 49 + 576 = 625, so OP = 25 cm. For chord AB and point C on the circle, choose tangent ray AX opposite C across line AB. The alternate segment theorem gives angle XAB = angle ACB. The other tangent ray gives the supplementary angle, totalling 180 degrees with angle XAB. To finish a triangle, subtract two known angles from 180. To split a tangent-radius right angle, subtract a known part from 90. Triangle angles 47 and 68 leave 65 degrees. Radii OA and OB make triangle OAB isosceles. Its base angles are equal: each is (180 - angle AOB) / 2. At A, radius OA is perpendicular to the tangent. If chord AB lies inside that right angle, subtract angle OAB from 90 to find the tangent-chord angle. For a centre angle of 110 degrees, each base angle is 35 degrees, leaving a tangent-chord angle of 55 degrees. Tangents PA and PB form quadrilateral OAPB with centre O. Its angles at A and B are each 90 degrees. A quadrilateral's interior angles total 360 degrees, so its interior angles AOB and APB total 180 degrees. Subtract the outside-point angle from 180 to find this centre angle. **Recap.** Mark the tangent right angle, match tangent pairs, copy alternate segment angles, then finish with sums. *Lesson, questions and flashcards written for this dataset.* ## 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)