Equality begins at one source point
If tangents from P touch at A and B, then PA=PB. The shared external point controls the equality.
Learning goals
- Prove equal tangents from an external point.
- Use equal tangents in length, angle and polygon problems.
Concept lesson
Right angles and RHS
OA is perpendicular to PA, OB to PB, OA=OB and OP is common; these facts prepare RHS congruence.
Theorem 10.2 proof
Triangles OAP and OBP are right triangles, OA=OB and OP is common. RHS congruence gives PA=PB by CPCT.
Tangent equation
If PA=2x+3 and PB=5x-12, equate them because both start at P; x=5 and each length is 13.
Pair by source point
Colour-code each source point, equate only its tangent pair, then solve the length or perimeter equation.
Derive angle symmetry
Use congruent triangles to establish equal half-angles, then combine them with the two contact right angles in quadrilateral OAPB.
Polygon tangent ledger
List equal tangent pairs vertex by vertex, rewrite each side as two pieces, and derive the required perimeter or opposite-side relation.
Integrated tangent chain
Combine perpendicular radii, RHS congruence, equal tangents, angle sum and algebra; write the reason beside each equality.
Proof-to-application ladder
Progress from stating equality, through its RHS proof, to equations, angles and circumscribed figures; match reasoning depth to marks.
External-tangent mastery
Prove equal tangents, solve a tangent equation, determine an angle and complete a polygon relation with reasons.