Chapter 10. Circles
4 sections · about 10 pages in the Mathematics book (jemh110.pdf). Section anchors below are the targets of the citation chips you get back from an answer.
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Concept animations for this chapter
Interactive explanations, drawn in the page — no video, no external scripts. Each one is keyboard operable, has a “what to notice” caption, and a “try this” prompt that changes what it shows.
Circles and tangents — tangent length from an external point
conceptId: circle-tangents
Tangent length from an external point
CBSE Class 10 Mathematics · Ch 10 §10.2 — a tangent touches the circle at exactly one point, and the radius to that point is perpendicular to it.
- tangent length PT = √(OP² − r²)
- 4.583
- OP² − r²
- 21.000
- secant PA · PB = (OP − r)(OP + r)
- 21.000
- ∠OPT (should be arcsin(r/OP))
- 23.58°
- OT · TP (perpendicular ⇒ 0)
- 4.44e-16
What to notice: the two tangent lengths from P are equal, and PT² is exactly OP² − r². Switch the secant on and the same number reappears as PA · PB — the tangent–secant relation, PT² = PA · PB.
Try this: set OP = 5 and r = 3 — PT comes out exactly 4, the 3-4-5 triangle. Then drag OP down to r and watch the tangent length shrink to 0 as P lands on the circle.
10 further animations in the registries belong to other chapters.