Understand tangent properties, the tangent-secant relationship, and how intersecting chords relate their segment lengths.
A tangent to a circle is perpendicular to the radius at the point of contact. Two tangents drawn from an external point are equal in length.
Two tangents are drawn from an external point P to a circle, touching at A and B. If PA=7 cm, find PB.
PB=7 cm, since tangents from an external point are equal
A tangent touches a circle at point A. What angle does it make with the radius OA?
90°, since a tangent is always perpendicular to the radius at the point of contact
When two chords intersect inside a circle, the products of their segments are equal: AP×PB=CP×PD.
Two chords intersect at P, with AP=4, PB=6, CP=3. Find PD.
4×6=3×PD; PD=24/3=8
Two chords intersect at P, with AP=5, PB=8, CP=4, PD=10. Verify the theorem.
5×8=40; 4×10=40; equal, confirming the theorem
1. Two tangents from an external point P touch a circle at A and B. If PA=9 cm, find PB.
2. Two chords intersect at P, with AP=6, PB=8, CP=4. Find PD.
3. A tangent PT and secant PAB are drawn from external point P. If PT²=PA×PB, and PA=4, PB=9, find PT.
4. Two chords intersect at P, with AP=3, PB=12, CP=6. Find PD.
5. What angle does a tangent make with the radius at the point of contact?