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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
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.
Mark the tangent right angle, match tangent pairs, copy alternate segment angles, then finish with sums.
2 · Watch
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.